Let $I$ be a ideal of $\Omega(M)$ ($M$ a manifold) that is locally generated by $p$ independent $1$-forms $\omega_1,...,\omega_p$. Set $\omega = \omega_1 \wedge ...\wedge \omega_p$ Then $I$ is a differential ideal iff $$d\omega = \alpha\wedge\omega $$for some $1$-form $\alpha$.
The $\implies$ direction is straightforward, since $d\omega_i = \sum \eta_j \wedge \omega_i$ for some $1$-forms $\eta$. The other direction has got me stuck. I thought about trying to prove that $d\omega_i \in I$, but I don't see how to use this hypothesis since it seems like I"m losing a lot of data.