Can every closed differential form be expressed via constant coefficients? Let $M$ be a smooth $n$ dimensional manifold, and let $1 \le k < n$. Let $\omega \in \Omega^k(M)$ be a closed $k$-form on $M$.

Let $p \in M$. Do there exist coordinates around $p$, such that $\omega=a_{i_1i_2\dots i_k}dx^{i_1} \wedge dx^{i_2} \dots \wedge dx^{i_k}$, where $a_{i_1i_2\dots i_k}$ are constants?

That is, I ask whether every closed differential form be locally expressed via constant coefficients. 
Edit:
I forgot to require that $\omega$ should be everywhere non-zero. Otherwise, as mentioned by Paulo Mourão, one can take $xdx$ on $M=\mathbb{R}$.

I believe you don't have such coordinates around $0$ for $\omega=xdx$ in $\mathbb{R}$.
