My law school had no problem with hats. Alot of students looked like they just fell out of bed. Though it seems when you're trying to hide something, that's when Murphy's law kick's in, so disregard what I typed earlier. You are too young and it sounds like your hair loss is aggressive. As for the ladies (though you say you don't have a problem), if your friends have their hair, bring your bank statement with you. J/K, women hate those type of guys. I would try to buzz it for a while. Invest that money! Law school is expensive.
Oh yea, run from hair mills.