It is known, that one can convert any function $f(x_1,\dots,x_n)$, defined near $0$, into the function $(y_1,\dots,y_n)\mapsto a+y_1$, by a suitable local change of coordinates, provided $df\neq 0$. It follows from implicit function theorem.
I would like to know if there is a similar simplest look for any $n$-form $\omega$, and what is this look.
For instance, is it true that if $d\omega\neq 0$, for $\omega$ a $k$-form, then $$\omega=x_{k+1}dx_1\dots d x_{k}, $$ modulo closed forms, for some change of variables. If so, how to show it?
What about closed forms, is there some standard look locally?
(I suspect that the answer may be different for closed and non-closed forms, so if there are some results only for some types of forms, I would also like to know it.)
It is not true that any non-closed $k$-form looks like $\omega = x_{k+1} dx_1\wedge\cdots\wedge dx_k$, modulo closed forms. For then you'd have $d\omega = dx_{k+1} \wedge dx_1 \wedge \cdots \wedge dx_k$. Then $(d\omega)^2 = 0$ but if $k$ is odd, in general you can find a closed $k$-form $\omega$ with $d\omega \wedge d\omega \ne 0$.
More generally, such a form as $\mu = dx_{k+1} \wedge dx_1\wedge\cdots\wedge dx_k$ has a very large nullspace when considered as a map $TM \to \Lambda^{k-1} T^*M, v \mapsto i_v \mu$, where $i_v$ denotes interior multiplication. If you restrict to prescribed rank then maybe something is possible. For example, Moser's theorem states that any closed non-degenerate two form on a $2n$-dimensional space is (locally) of the form $dx_1 \wedge dx_2 + \cdots + dx_{2n-1} \wedge dx_{2n}$. However, the common proof (which is very slick and pretty, and I recommend looking up) is pretty unique to 2-forms.