Then, how can we show that $\forall i,j\in \mathbb Z, s.t.:1\leq i<j\leq n$, $e_i\wedge e_j $ is a basis vector for $\wedge^2(V) $?

Question

Let $V$ be a n dimensional vector space. Suppose $x,y\in V, f,g\in V^*$. Define $f\wedge g(x,y) = det \left( {\begin{array}{cc} fx & fy \\ gx & gy \\ \end{array} } \right)$

Then, how can we show that $\forall i,j\in \mathbb Z, s.t.:1\leq i<j\leq n$, $e_i\wedge e_j $ is a basis vector for $\wedge^2(V) $? ($\wedge^2 (V)$ is the set of antisymmetric 2 forms on $V$, and it is also a vector space).

Answer 1

Let $\omega$ be a 2 form. You need to write is as a linear combination of $e_i \wedge e_j$. What is your guess for, say, the coefficient of $e_1 \wedge e_2$?