Normal forms for exterior forms is a classical subject. It is well-known that when $k=0,1,2,n-2,n-1,n$$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'normal forms', i. Seee., 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, for example. 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.