How to Perform Skullcrushers
Bodybuilders looking to naturally increase their arm size as rapidly as possible should focus on compound and isolation movements targeting the triceps, rather than the biceps. Powerlifters who grind through the midpoint of or fail to lockout their bench press reps should hammer their triceps with compound movements.
The triceps are primarily built through pushing movements such as presses and extensions. These movements may be performed in a vertical or horizontal plane. Every balanced physique needs a beastly set of triceps.
The barbell skullcrusher is an isolation push exercise targeting the triceps brachii, comprised of the long, lateral, and medial heads. There are no supporting muscles groups to assist the target muscle group during this movement. However the anterior or front deltoid, clavicular and sternal heads of the pectoralis major, latissimus dorsi, teres major (outer back), posterior or rear deltoid, and wrist flexors act as stabilizers during this exercise. 
Stabilizer muscles help maintain a posture or fixate a joint by contracting without significantly moving.  Skullcrushers are an excellent lift for building tricep size and strength but they may exacerbate preexisting elbow issues.
MTS Nutrition CEO Marc Lobliner discusses proper skullcrushers form.
How to Perform the Skullcrusher
Place a short straight barbell or EZ bar on the floor and select the appropriate working weight. Ensure you add an even amount of weight to both sides of the bar. don't place 10lbs on one sides and 25lbs on the other side. Uneven loading won't improve your gains and will likely lead to an injury.
If this is your first time performing the exercise then pick a conservative weight that you can safely lift for 8 to 12 repetitions. When in doubt, start with a conservative weight and work your way up; this will leave more room for progression and ensure you develop properly form early on.
Once you've selected the appropriate working weight grasp the barbell with a pronated grip (palms facing away from you) slightly narrower than shoulder width. Taking a grip outside of shoulder width will decrease range of motion and cause the elbows to drift away from the torso which minimizes the stimulus of the triceps.
An excessively narrow grip will lead to excessive internal rotation of the shoulders and may exacerbate preexisting elbow issues even further. You can use a traditional grip (thumb wrapped around the fingers) or a false grip (thumb and fingers on the same side of the handle). Lay down on a flat bench so that your glutes, back, and head are in contact with the padded support. Ensure your chest is high, upper back is squeezed, and shoulder blades are retracted. Your feet should be flat and your shins should be perpendicular to the ground.
Now that you're lying face-up on the bench straighten (but do not hyperextend) your arms and elbows so that the barbell is vertically in-line with your shoulders. This is your starting position. Take a deep breath, brace your abdominals for impact and begin lowering the barbell to your forehead.
Initiate this lowering by bending your elbows. Your upper arm and elbows should remain in a fixed position close to the torso but your forearms will move towards your forehead. Continue lowering the barbell until it's only an inch or two from your forehead.
Some trainees find that lowering the barbell to the top of or behind the head rather than the forehead decreases some of the stress on the elbows. Both variations are acceptable; experiment with both and see which feels most comfortable for you.
Hold this bottom, stretched position for 1 to 5 seconds. Initiate the return to the starting position by flexing the triceps and pushing the barbell up and back to the starting position. Complete for the desired number of repetitions. The movement path for lowering and raising the barbell should be the same. If your elbows drift or arms flare outwards during any portion of this lift then the weight is too heavy.
Some lifters choose to exhale while extending elbows and moving the barbell upwards while others prefer breathing in between each repetition. Choose a breathing pattern that feels the most natural and comfortable for you.
This exercise can be performed using straight sets, pre-exhaust sets, drop sets, rest-pause sets, supersets, trisets, giant sets, paused reps, partial reps, forced reps, or slow negatives. As with any exercise, the two most important components are high-quality form and progression. Progression can take a variety of forms (e.g. more weight, sets, or reps, decreased rest period, improved rep quality, etc...) but strive to improve every training session.
Skullcrusher Form TipsMinimize Elbow Flaring - The barbell skullcrusher may cause elbow discomfort in even the healthiest trainees. In addition to using conservative working weights on this exercise ensure your upper arm and elbows stay close to the torso throughout the entire movement.
If you find yourself flaring your elbows and upper arms to complete the movement then the weight is too heavy. Keeping your elbows in-line with your shoulders will ensure your triceps, rather than your elbows, are stressed.
Avoid Momentum - The barbell skullcrusher provides maximum benefits when it's perform in a controlled full-range of motion. Check your ego at the door and don't immediately attempt 100lbs. In addition to staying tight (glutes squeezed and shoulder blades retracted), keep your upper back on the padded support.
Do not use momentum or raise your hips off the pad to bounce the barbell from the bottom position to the starting position. This momentum dramatically increases the likelihood of injury and minimizes the stimulus of the target muscles.
Hold the Stretch - If you're looking to increase intensity then experiment with holding the bottom, stretched position of the barbell skullcrusher for 5 to 10 seconds. Really focus on squeezing the triceps. This will increase time under tension and the burn in the triceps. Increased time under tension is an excellent variable to adjust for progressive overload and enhanced muscle growth.
References1) "Barbell Lying Triceps Extension "Skull Crusher"." ExRx (Exercise Prescription) on the Internet. N.p., n.d. Web. 5 Nov. 2015.
2) "Kinesiology Glossary." ExRx (Exercise Prescription) on the Internet. N.p., n.d. Web. 5 Nov. 2015.