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

Thanks in advance!