Using the fact that the quotient map from S^n to RP^n (x-->[x], where x~y iff y=±x) is a local diffeomorphism and the fact that the antipodal map above is orientation reversing, I can get a contradiction from the hypotheses that RP^n is orientable when n is even. But when n is odd, it's another game. How does that work?

When n is odd, the quotient map f:S^n->RP^n is locally a diffeomorphism, so it locally maps an orientation on S^n to RP^n and, the fact that x->-x is orientation preserving means that the orientation given by antipodal points agree, giving a globally defined orientation on RP^n.

It's not very elegant though... I start by considering an atlas of consistently oriented charts of S^n such that
(1) The restriction of p to each coordinate chart is a diffeomorphism onto its image.
(2) if U is a coordinate chart around x and V a coordinate chart around -x, then U and V do not intersect.

I then push the atlas to RP^n via p, and now it remains to show that this atlas of RP^n is too, consistently oriented. So I take some point [x] in RP^n. Then for any chart U'=p(U), V'=p(V) around [x], either
(a) U n V is empty and (W.L.O.G.) x is in U, -x is in V
(b) U and V intersect

In each case, the transition function from U' to V' is shown to have positive determinant. In case (a), the antipodal map comes in and the fact that it itself has positive determinant (for n odd) is the key.