The wikipedia page on exterior algebras makes the following reasonable sounding statement (I paraphrase):
Let $V$ be a complex vector space and consider the second exterior power $\bigwedge^2 V$. By the rank of $\alpha \in V$, we mean the smallest number $p$ such that $\alpha$ can be written as a sum of $p$ decomposable elements. CLAIM: $\alpha$ has rank $p$ if and only if the $p$-fold wedge product $\alpha \wedge \dots \wedge \alpha$ is nonzero but the $(p+1)$-fold wedge product is 0.
Unfortunately I can't figure out a proof of this, even in the case $p=1$, though it feels like it ought to be elementary. I've looked in a bunch of algebra books, but none of them explain it, probably because it is really easy and I am just missing something.