Consider the following chain $\{A\_1,A\_2,A\_3,\cdots,A\_{n}\}$ of orbit spaces of even-rank anti-symmetric tensors, where
$$A\_k:=\frac{\Lambda^{2k}(\mathbb{R}^{2n})}{e\_{i\_1}\wedge \cdots \wedge e\_{i\_{2k}}\mapsto Re\_{i\_1}\wedge \cdots \wedge Re\_{i\_{2k}}},~~~~~~~~~R\in O(2n),$$
The base case is well-known, and corresponds to the canonical form for a real, antisymmetric matrix:
$$A\_1=\frac{\Lambda^2(\mathbb{R}^{2n})}{e\_i\wedge e\_j\mapsto Re\_i\wedge Re\_j}=\left\{\bigg[\sum\_{j}\lambda\_{j} e\_{2j-1}\wedge e\_{2j}\bigg],~~\lambda\_j\in\mathbb{R}\right\}\simeq \mathbb{R}^n$$
Are there any results for $A\_2$?



Classification of $2k$-vectors modulo orthogonal transformations
Actually, there is a fair amount known in the first nontrivial case: $(k,n) = (2,4)$. For example, see *Calibrations on $R^8$* by J. Dadok, R. Harvey and F. Morgan Transactions of the American Mathematical Society Vol. 307, No. 1 (May, 1988), pp. 1-40. 


Also, see Antonyan, L. V.
*Classification of four-vectors of an eight-dimensional space.* (Russian) 
Trudy Sem. Vektor. Tenzor. Anal. No. 20 (1981), 144–161. 


Already for higher values $(k,n) = (2,n)$ for $n>4$, I think very little is known.


