I was reading this question, and the first answer says:
There is a canonical form: there is a basis $e_1,\cdots,e_n$ of $V$ and a $k$ such that $w=e_1 \wedge e_2 + \cdots + e_{2k−1} \wedge e_{2k}$.
Why is this true? I wasn't able to come up with a proof. My knowledge of exterior algebra is the most basic. If I understand correctly, vectors $e_{i_1} \wedge e_{i_2}$ form the basis of $\bigwedge^2(V)$, but wouldn't that mean that every $w \in \bigwedge^2(V)$ could be written as $a_1(e_1 \wedge e_2) + a_2(e_1 \wedge e_3) + \cdots$, meaning that not only $e_k \wedge e_{k + 1}$ terms are present, but also $e_k \wedge e_{k + p}$? I'm very confused about this.