Why does my chicken get dry skin in the freezer?

The name for what you're seeing is freezer burn. It happens because water sublimates out of the exposed part of the meat while in the freezer, leaving it all dried out and unpleasant looking. To solve the problem, wrap your chicken (or anything else) tightly in plastic wrap, or place in freezer bags and press out all the air before sealing, and then freeze.

I've actually found a combination of both aluminum foil and clingfilm to give the best protection. The foil is most effective at preventing freezer burn, but does nothing to isolate odors.

Since the freezer is so cold, you don't ordinarily smell much when you stick your head in there, but that doesn't mean that the odors don't spread about, and you'll notice it only once you thaw your chicken (or other food item). That's why I like to wrap my food in clingfilm as well, but more pertinent to your question, the foil is what you want to prevent freezer burn.

I do this with my chicken every time. Buy the bulk pack and portion out at home. I portion into zipper top bags and lay the chicken inside neatly (single layer, flat as possible) and manually squeeze out all the air before zipping the top. I am a little obsessive about trying to squeeze every last pocket of air out, but am too lazy/cheap to get out my vacuum sealer for chicken (save those bags for more expensive meats). I defrost/use the chicken from 1-12 weeks later and haven't had any freezer burn in years.

Applying cling wrap should work. If the problems continue, then you might want to invest in a vacuum sealer.

I like Caleb's answer but I'd just like to add some detail (and I guess get some karma :P)

Caleb's suggestion--wrapping tightly in plastic wrap, is probably a good enough solution if you're using the chicken soon enough. The comment discussion on his answer is somewhat correct--most modern plastic wrap will not prevent freezer burn forever. The formulas for plastic wrap have changed over time and have different molecule permeability protections--some will allow water molecules through (slowly) and some won't.

Your chicken skin is drying out to sublimation--ice converting to water vapor without becoming a liquid first. As I understand it, this is a function of humidity, temperature and pressure. If you had a perfect seal--you'd still get a very small amount of sublimation, but the humidity inside your sealed bag/container would quickly rise to the saturation point. Since the water vapor can't escape, the bag consistently has a high humidity inside and this prevents/minimizes further sublimation.

What vacuum sealers give you is a sealed container, but they're not the only way to do that. I suspect that if you were to freeze a piece of chicken fully immersed in water, it would preserve similarly to a vacuum frozen one.