Rank of a 2-vector (exterior algebra)

[jojo12345]

I understand that there is a way to find a basis $$\{e_1,...,e_n\}$$ of a vector space $$V$$ such that a 2-vector $$A$$ can be expressed as

$$A = e_1\wedge e_2 + e_3\wedge e_4 + ...+e_{2r-1}\wedge e_{2r}$$

where 2r is denoted as the rank of $$A$$. However the way that I know to prove this seems sort of inelegant. I'm wondering what other proofs people have.

 

[jojo12345]

I'm especially curious if there is a geometric interpretation of the result.