Rank of a 2-vector (exterior algebra)

jojo12345
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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.
 
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I'm especially curious if there is a geometric interpretation of the result.
 

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