"Hair-brained" is an accepted alternative. Two theories about what it means, to go with the two spellings.

"Hare-brained" is because you act like a March Hare, when they get a bit over-lively,or at least the gentlemen-hares do, because they are trying to impress the lady-hares.

"Hair-brained" would be acting as if your brains had been replaced by hair, like a stuffed horsehair cushion.