let p:S^n \to P\mathbb{R}^n be the map that is obtained by restricting the canonical projection \mathbb{R}^{n+1} \to  P\mathbb{R}^n to S^n.
This is a covering map of manifolds. And now I want to use this fact to show that P\mathbb{R}^n is orientable for n odd and non orientable for n even.
It looks like I have to use that the tangent bundle of S^n is trivial if and only if n is odd, but I don't see the trick.

Thanks in advance!