I am working on the following problem.
Let $M$ be a smooth orientable $n$-manifold, $n \geq 2$, and let $\varphi$ be a smooth $(n-1)$-form on $M$. Show that there is a vector field $v$ on $M$ such that $$\varphi(v,\ \_\ , \dots, \_\ ) \equiv 0$$ and $v \neq 0$ at every point where $\varphi \neq 0$.
The condition can be rewritten as $i_v\varphi = 0$.
If I could show that $\varphi = i_w\omega$ where $w$ is a vector field and $\omega$ is an orientation form on $M$, then I could take $v = w$. Is it the case that every $(n-1)$-form on a smooth orientable $n$-manifold can be written in this way?
If this is not the case, I'd appreciate some hints on how to proceed.