Let $M = (E,\mathcal I)$ be a linear matroid represented by $\rho: E \rightarrow V$. This induces a map $$ \hat\rho: \mathcal I \rightarrow \mathbf P(\Lambda V), \{e_1,\ldots,e_k\} \mapsto [\rho(e_1) \wedge \cdots \wedge \rho(e_k)]. $$ The subset $\operatorname{Im}(\hat\rho) \subset \mathbf P(\Lambda V)$ in turn defines a linear matroid $\hat M$ over $\Lambda V$.

Is there anything known about this matroid $\hat M$, e.g. how it depends on the choice of representation, and what its bases look like in relation to the bases of $M$? I'm interested in particular in the case where $M$ is uniform (for instance, I think the representation shouldn't matter in that case) or arises combinatorially, and would like to obtain combinatorial descriptions of $\hat M$. But any pointers would be greatly appreciated.