Given a nowhere-zero, closed $2n$-form $\Omega$ in a manifold of dimension $2n +1$, how do we know if there exists a closed $2$-form $\omega$ such that $\Omega = \omega^n$?

Remark. This question started off as a question on multilinear algebra because I thought that perhaps there was an algebraic point-wise condition on $\Omega$ that was non-trivial, but that is not the case: every element of $\Lambda^{2n}(\mathbb{R}^{2n+1})$ is the $n$-th power of an element of $\Lambda^{2}(\mathbb{R}^{2n+1})$.

Background. The odd dimensional manifolds in which I'm really interested are spherized tangent bundles (i.e. $STM := (TM \setminus 0)/\mathbb{R}^+$. The reason for this is that this question comes up in the study of inverse problems in the calculus of variations.