Let us start with the following definition.


Let $1\leqslant k\leqslant n$ and let $\omega\_1,\omega\_2\in\Lambda^k(\mathbb{R}^n)$. We say that $\omega\_1$, $\omega\_2$ are **equivalent**, if there exists $T\in GL(n,\mathbb{R})$ such that 
$$
T^\ast\omega\_1=\omega\_2,
$$
where $T^\ast:\Lambda^k(\mathbb{R}^n)\rightarrow\Lambda^k(\mathbb{R}^n)$ is the standard lift of $T$ to $\Lambda^k(\mathbb{R}^n)$.


It is known that:


1) When $k=1$, if $\omega\_1,\omega\_2$ are equivalent if and only if either $\omega\_1,\omega\_2\neq 0$ or $\omega\_1=\omega\_2=0.$ This is trivial to check.


2) When $k=2$, $\omega\_1,\omega\_2$ are equivalent if and only if, there exists $r\in\mathbb{N}$ such that 
$$
\omega\_1^r,\omega\_2^r\neq 0,\text{ and }\omega\_1^{r+1}=\omega\_2^{r+1}=0,
$$
where power corresponds to the wedge power. A proof of this result can be found in Denis Serres' book (Matrices: Theory and Applications). 


*QUESTION:* 


Is there any such result when $k\geqslant 3$? Probably $k=n-1,n$ cases are easy to handle. What I'm curious to know if there are analogous results when $3\leqslant k\leqslant n-2$?



Equivalence of exterior forms
Normal forms for exterior forms is a classical subject. It is well-known that when $k=1,2,n{-}2,n{-}1$, or $n$ and when $(n,k)=(6,3),(7,3),(7,4),(8,3),(8,5)$, there are only a finite number of 'normal forms', i.e., orbits in $\Lambda^k(\mathbb{R}^n)$ under the action of $\mathrm{GL}(n,\mathbb{R})$. 


In particular, in these cases, the 'generic' orbit is an open subset of $\Lambda^k(\mathbb{R}^n)$ and these orbits (which are finite in number) are said to be *stable*. In the first set of cases (which the OP briefly discussed above), there is usually only one open orbit, though, in the case $(n,k) = (4m+2,4m)$ when $m>0$, it turns out that there are $2$ open orbits.


For the five 'exceptional cases' listed above, one could consult Nigel Hitchin's discussion of *stable forms*. In these cases, typically there are $2$ open orbits, but in the cases $(n,k)=(8,3)$ or $(8,5)$, there are $3$ open orbits.


In all other cases, there are continuous moduli.


