Let $M$ be a smooth $n$ dimensional manifold, and let $1 \le k < n$. Let $\alpha $ be a closed locally decomposable non-zero $k$-form on $M$.
By decomposable I mean that $\alpha$ be can written (locally) as $\alpha=\alpha_1 \wedge \dots \wedge \alpha_k$, where $\alpha_i $ are smooth $1$-forms.
Let $p \in M$. How to prove there exist coordinates $x_1,\dots,x_n$ around $p$ such that $\alpha = \mathrm{d}x^1\wedge\cdots\wedge\mathrm{d}x^k$?
I am trying to understand first the easier case when we assume the $\alpha_i$ are closed:
Then locally $\alpha_i=df_i$, where the $f_i$ are some smooth functions defined on an open subset $U \subseteq M$. If we could complete the $\alpha_i$ to a local frame of closed one-forms $\alpha_1, \dots, \alpha_n$, we would have $\alpha_i=df_i$ for every $1 \le i \le n$ would be linearly independent. Thus the map $U \to \mathbb{R}^d$, given by $x \to (f_1(x),\dots,f_n(x))$ would be a local diffeomorphism (by the inverse function theorem, since all the $df_i$ are independent). This means the $f_i$ are the required coordinates.