On the identification of $\mathbb R^4\wedge\mathbb R^4$ with the set of antisymmetric matrixes in $\mathbb R^{4\times 4}$

Question

In view of the ususal definition of the electromagnetic field tensor, I assume that $\mathbb R^4\wedge\mathbb R^4$ can be identified with the subset of $\mathbb R^{4\times 4}$ consisting of antisymmetric matrices. Can someone tell me the precise form of this isomorphism$$\mathbb R^4\wedge\mathbb R^4\to\mathbb R^{4\times 4}_{\mathrm{antisymmetric}}$$i.e. can someone either explain to what matrix $x\wedge y$ is mapped for $x,y\in\mathbb R^4$ or to what matrices the elements of the standard basis are mapped?

Answer 1

A basis of $\mathbb{R}^4\wedge\mathbb{R}^4$ is $$\{e_1\wedge e_2,e_2\wedge e_3,e_3\wedge e_4, e_4\wedge e_1, e_2\wedge e_4, e_1\wedge e_3\}. $$ You can associated to $e_i\wedge e_j$ the matrix with $1$ on position $(i,j)$, $-1$ on position $(j,i)$ and $0$ elsewhere.