Consider the map from $M(n, \mathbb Z) \rightarrow M(m, \mathbb Z)$$M(n, \mathbb Z) \rightarrow M(\binom{n}{2}, \mathbb Z)$ taking a matrix A to its second compound, i.e, $\bigwedge^2 A$. Restricting this map to the invertible matrices we get a homomorphism of groups from $GL(n, \mathbb Z)$$\mathrm{GL}(n, \mathbb Z)$ to $GL(m, \mathbb Z)$$\mathrm{GL}(\binom{n}{2}, \mathbb Z)$.
Q1. How can we determine if a given matrix $B \in GL(m, \mathbb Z)$ is contained in the image of this map?
How can we determine if a given matrix $B \in \mathrm{GL}(\binom{n}{2}, \mathbb Z)$ is contained in the image of this map?
