Given a $2n$-form $\Omega$ in $\mathbb{R}^{2n+1}$, how do we know if there exists a $2$-form $\omega$ such that $\Omega = \omega^n$?

I would really like something very explicit. For example, if $$ \Omega = \sum_i A_i \ dx_1 \wedge \cdots \wedge \hat{dx_i} \wedge \cdots \wedge dx_{2n + 1} \ , $$ then what is the condition on the numbers $A_i$ which are necessary and sufficient for the existence of a two form $\omega$ satisfying $\Omega = \omega^n$.

I feel this should be known (something neat such as Cartan's characterization of simple $k$-forms), so this may be more of a reference request. I would also be interested in knowing whether there is a way to know whether $\Omega$ is the product of $n$ (possibly different) $2$-forms.