If $\omega$ is a differential $4$-form on a $10$-manifold $M$ then $\omega \wedge d\omega$ is exact

differential-formsdifferential-geometrymanifoldssmooth-manifolds

Let $\omega$ be a differential $4$-form on a $10$-manifold $M$. I am trying to show that $\omega \wedge d\omega $, which is a $9$-form, is exact.

Clearly $\omega \wedge d\omega$ is closed, because $d\omega \wedge d\omega =0$ ($|d\omega|=5$ is odd). But how can we show that $\omega \wedge d\omega $ is exact?