I've been stuck on this problem from Katok & Hasselblat for a while. I think it shouldn't be hard but I can't make progress.
Question. Suppose $M$ is a non-orientable manifold and $\pi:M^*\rightarrow M$ is an oriented double cover, and $I: M^* \rightarrow M^*$ is the corresponding involution. Prove that there exists a non-degenerate $n$-form such that $I^*\omega = -\omega$ and hence $\pi_*|\omega|$ is an odd $n$-form on $M$.
I think what they mean by the involution is the map
$$I\left(p, \pm\left[\left.\partial/\partial x^1\right|_p, ..., \left.\partial/\partial x^1\right|_p\right]\right) := \left(p, \mp\left[\left.\partial/\partial x^1\right|_p, ..., \left.\partial/\partial x^1\right|_p\right]\right)$$ where $x: U\rightarrow \mathcal{O} \subseteq \mathbb{R}^n$ is a chart about $p$ on $M$.
What I think is that since $M^*$ is orientable, one should be able to take a volume form $\Omega$ on $M^*$ and then define $\omega:= \Omega - I^*\Omega$. Obviously $I^*\Omega = - \Omega$. However, $\omega$ may fail to be non-degenerate. This would mean, for some $p \in M$ and some chart $x$ about it, $$\omega_{\left(p, \left[\left.\partial/\partial x^1\right|_p, ..., \left.\partial/\partial x^1\right|_p\right]\right)} = \left(I^*w\right)_{\left(p, \left[\left.\partial/\partial x^1\right|_p, ..., \left.\partial/\partial x^1\right|_p\right]\right)}.$$
So far I wasn't able to rule this out. Indeed I suspect this might not work out.