I have the following short argument which seems to say that there are no non-vanishing, (n-1)-forms on a closed manifold of dimension n but I am not very confident in my understanding of a "volume form" because I am used to only working with $\textit{the}$ volume form (induced by a metric) so I don't know if this applies:
Suppose that $M$ is an oriented, n-dimensional manifold which is also closed (i.e. $\partial M$ is empty). Suppose also that $\alpha$ is a non-vanishing (n-1)-form. Then $d\alpha$ is an n-form.
$\textbf{Question 1:}$ Does that mean that $d\alpha$ is a "volume form"? Can I conclude that $\int_{M}d\alpha$ is nonzero? Let's suppose the answer is yes. Let $d\alpha = Vol_{\alpha}$.
$\textbf{Question 2:}$ In this case, it seems like Stokes theorem and the fact that $M$ is closed says:
$$0 \neq \int_{M}Vol_{\alpha} = \int_{M}d \alpha = \int_{\partial M} \alpha = 0$$
Which is a contradiction.
If this fails, where does it fail? Are there any assumptions I can add to $\alpha$ so that it would work?