Old Standby 2: is orientable iff is odd.
First note that the antipodal map by is orientation preserving if n is odd and orientation reversing if n is even just because in coordinates it is the matrix with ‘s on the diagonal.
Now if we make our natural identifications between as a manifold and as the vector space that is tangent space at a given point, then we see that if we restrict to embedded in the orientation preserving/reversing still holds. This is just because if is an oriented basis at , then is an oriented basis at that same point in . Thus the orientation at of is .
Now suppose that n is even and that has an orientation. Let be the standard quotient map. But now the orientation of induces an orientation on by letting an ordered basis be positively oriented if is positively oriented.
But the induced map of on is just the identity. Thus is orientation preserving on , a contradiction.
The other direction we need to put an orientation on by . Suppose n is odd now. Define a basis to be positively if there exists a positively oriented basis such that . We just need to make sure this is a well-defined choice. But it is since the fibers of are and , and we get from one to the other through the antipodal map which is orientation preserving. So we’re done.
Let’s do some analysis of this. First off, there was nothing special about this particular . So we actually proved that if is a smooth covering and is orientable, then is also orientable.
I know of two other ways to prove this. Both require that antipodal map observation first. One way is to prove the more general fact that if is a connected, oriented, smooth manifold and is a discrete group acting smoothly, freely, and properly on M, then is orientable iff is an orientation preserving diffeo for all . In this case, and the 1 action is the identity and the action is the antipodal map.
The other way is far more elegant. It is the algebraic topology method (actually I believe you can prove the stronger statement of the second method using this way). I haven’t quite reworked this way out, yet, but it just involves pulling back nowhere vanishing n-forms or something.