There is a natural identification $\Lambda^2V\cong\mathfrak{so}(V)$ where $a\wedge b$ acts as the right-angle plane rotation in the oriented plane spanned by an ordered orthonormal set $\{a,b\}$. A version of the spectral theorem leads us to conclude that every element of $\Lambda^2V$ is expressible in a "canonical form"
$$ x=a_1\wedge b_1+\cdots+a_n\wedge b_n \tag{$\star$}$$
with $\{a_1,b_1,\cdots,a_n,b_n\}$ orthogonal and $|a_i|=|b_i|$ for each $i$. There may be redundancy: for instance if all of the vectors have the same size, then the set of real 2D planes they span may be chosen to be the 1D complex spans of any orthonromal basis, treating $\mathrm{span}\{a_i,b_i\}$ as a complex vector space (and using $x$ for multiplication-by-$i$). Indeed, this explains all possible redundacy.
Problem One. Given any $a_1\wedge b_1+\cdots+a_n\wedge b_n$ (without assuming anything about $\{a_i,b_i\}$), how can we bring it into the form $(\star)$ using only the distrbutive and alternating properties of $\wedge$ and decomposing vectors into parallel/perpendicular components with respect to spans of other vectors already written out? (So, a coordinate-free procedure. In particular, we are avoiding rewriting them as matrices and using matrix decompositions.)
The simplest case would be with a pair of wedges, without loss of generality $a\wedge b+c\wedge d$ where the vectors $a,b,c,d$ are linearly independent in $\mathbb{R}^4$. One can also without loss of generality assume that $a,b,c$ are orthogonal and $d$ is orthogonal to $c$, but going from there seems to increase the number of wedges being added. How to get cancellation towards the desired form?
Problem Two. Is there some kind of canonical form for elements of $\Lambda^rV$ in general?
- Presumably such a canonical form would involve expression as a sum of pure wedges associated to "semiperpendicular" subspaces (i.e. giveny two subspaces $V$ and $W$, the orthogonal complement of $V\cap W$ within $V$ and $W$ are orthogonal within $V+W$). This is a fairly week observation though; just picking a coordinate basis tells us it's possible.
- Given any such canonical form, if the spans associated to the pure wedges are $A_1,\cdots,A_k$, we get a lattice map $\mathcal{P}(\{1,\cdots,k\})\to\{1,\cdots,r\}$ given by $S\mapsto \dim\bigcap_{s\in S}A_s$; is this "dimension signature" of the canonical form uniquely determined by the $x\in\Lambda^rV$?
- What kind of redundancy can we expect in the canonical form of such an element $x$?
- Does the algorithm in Problem One generalize to $r>2$?