Suppose $ A $ is a skew-symmetric matrix with elements in $ \mathbb{R}. $
Show that it exists an Unique linear map $ L$ such that
$$ L: \bigwedge^2 \mathbb{C}^3 \to \mathbb{C} \\ (x \wedge y) \mapsto x^T Ay \quad \forall x,y \in \mathbb{C}^3. $$
Attempt for solution;
$ ( x \wedge y) $ is bilinear and alternating. Thus, $ (x \wedge y ) = - ( y \wedge x) $, and $ L(x \wedge y) = L(-y \wedge x) $.
I dont know how to show it's an unique map. Does the linearity imply uniqness?