We can do it in Fallout 3 (in your home), so I think it's odd that we can't do it in Skyrim... Is there any way to change your character's hairstyle?

If you are on PC just type showracemenu into the console and you can change the warpaint and hairstyle situation. I think you are stuck if you are on xbox, unless there is DLC or a patch that adds the function.

Thank you!.

I am on PC, however I've tried to change my hairstyle (my hair only) using your system and I have found that ... I lose my racial bonuses!!

Yeah. That happens.
There is a way to pass it, though. Instead of choosing OK when you finish with your character's hirstyle, you should create a new save while in the character menu and load it. This way only the characters looks will change, not the skills/bonuses.
It worked in Oblivion, so I think it should work in Skyrim too.

Oh Snap! Sorry dude I didn't know that would happen!

I tried to do it, but the game not let me save when the character editor is opened

EDIT: seems like restarting the game restores the racial bonuses