The matroid you're considering is the direct sum of the matroids where you take the $k$th wedge power, so it suffices to study those for each $k$.

The question of whether, for an abstract matroid $M$, there is a matroid $\hat{M}$ with the properties that you expect has been studied in the paper "Glueing matroids together: a study of Dilworth truncations and matroid analogues of exterior and symmetric powers" by Mason. I'm not sure if there is a counterexample there (as the paper is hard to access), but one expects the answer to be negative, as there is a counterexample to the analogous question for tensor products in the paper ["On products of matroids"][1] by Las Vergnas. 

Brakensiek, Dhar, Gao, Gopi and I have [shown][2] that if the field has characteristic $0$ and $\rho$ is generic, then the dual of the matroid for $\wedge^2$ is Kalai's [hyperconnectivity][3] matroid. This matroid has been studied a fair amount, but in general there is nothing you could call a combinatorial description of it. 

Here you really need $\rho$ to be generic, representing the uniform matroid is *not* enough: there are realizations of the uniform matroid in characteristic $0$ for which $\wedge^2$ is not dual to the hyperconnectivity matroid. If $\rho$ is generic and the field has characteristic $p$, then I don't know if you still get the dual to the hyperconnectivity matroid.


  [1]: https://www.sciencedirect.com/science/article/pii/0012365X81901722
  [2]: https://arxiv.org/abs/2405.00778
  [3]: https://link.springer.com/content/pdf/10.1007/BF02582930.pdf?pdf=button