So it is well known that given a smooth vector field $X$ on a smooth manifold $M$, at each point $p\in M$ we can find coordinates $(U,x)$ such that $X = \frac{\partial}{\partial x^1}$ in these coordinates.
My question is, does there exist a similar statement for $1$-forms? i.e, given a one form $\omega \in \Omega^1(M)$, does there exist a chart $(U,x)$ around $p$ so that $\omega = dx^1$.
At the moment I'm unsure but I'm leaning towards it not being true in general.
So far I have that if $\ker\omega$ is integrable locally around $p$, then I feel this is sufficient for the existence of such a chart, since we can just take the frobenius chart and scale it as necessary. However, a counterexample or proof for the general case eludes me.
Any help to clear this up for me is greatly appreciated.
If the form is not closed, then it cannot be of that form (and if it is closed, then its kernel is involutive and you can use Frobenius's theorem.)