At the end of the day, the only thing that seems to set races apart in Skyrim is their racial abilities. There are no rigid classes, so play as you like (as other posters advised)
Relative skill bonuses exist, but are only relevant at the start... any race can achieve max lvl100 in all skills, and no-one does it faster than any of the others. Different races merely start out with their skills at different levels.
Thus, playing a high-elf is appropriate as long as you use a fair amount of magic - this allows you to benefit from their racial ability: faster magicka regeneration. The high-elf can learn the battle skills you desire as well as any other race.
A Dunmer (Dark Elf) is appropriate as a battle mage as well. He has the nice ancestral flames ability (if I remember correctly) which will damage enemies within a small radius from the player for 60s, as well as a 30% resistance to fire (could be useful in some fights?). They also start off with a well-mixed set of skill bonuses that lend themselves well to a battle mage.
However, if you've started with your high-elf already and like the character, stick to it. Only if you truly want to have the ancestral flames and fire resistance abilities (or a different look) would it be worth switching now. Skill bonuses should not be a primary concern.
I hope that was helpful, and apologise if I made any mistakes! I'm an ES noob as well and haven't even got the game yet... but I'm doing my research until my semester ends so that I can plunge into Skyrim well prepared

Cheers!