Eh, not exactly.
Your starting skills are a bit different. You could certainly have a high elf warrior, but at the same level his skill levels would be somewhat lower, which will also affect what perks you can choose.
The difference is that each class has a different racial power as well. Orcs get berserk, which makes them beastly in melee for about 30 seconds a day, and Nords are naturally resistant to frost and can perform a battle cry that makes all but the most resolute run like sissies.
High elves get a racial power that allows them to regenerate magicka faster for a short period each day, and I don't think they have a magicka weakness any more (which always made them unattractive to me in Morrowind). But, their starting skills give them starting bonuses to magic, not combat.
So, because Skyrim has done away with classes, you really have the freedom to do whatever you want! A High Elven warrior? Sure! He might not be as tough as a Nord at first, but give him some time and practice, and he will do alright!