Let $M$ be a $n$-dimensional manifold and $\alpha$ a smooth $r$-form in $M$. It is clear that $\alpha$ can be written locally as $\alpha=\alpha_1\wedge \ldots \wedge \alpha_r$ for some $1$-forms $\alpha_1,\ldots,\alpha_r$ just by considering its expression in a local chart of $M$.
In general, can we have this descomposition $\alpha=\alpha_1\wedge \ldots \wedge \alpha_r$ in the whole manifold? Maybe with the use of bump functions?
In looking for the easiest counter example, my initial answer was incorrect. However, it is still not true, even locally. For example, if we take $dx\wedge dy+dz\wedge dw$ on $\mathbb{R}^4$, you cannot write this as a wedge of $1$-forms, but trying to is probably a good exercise.