Do there exist polynomial relations which characterize the $k$-minors of a square $d \times d$ matrix over $\mathbb{R}$?

I am interested in the case when $k<d-1$.

It seems that finding explicit relations is an open problem, e.g. see this paper. However, I am not asking about an explicit construction, only on mere existence.

In fact, I would also be interested in relations expressed via arbitrary continuous functions, not necessarily polynomial.

To my understanding, this is equivalent to the assertion that the image of the following map is closed*:

$$ \psi:\text{Hom}(V,V) \to \text{Hom}(\bigwedge^kV,\bigwedge^kV) \, \, \, \, , \, \, \,\psi(A)=\bigwedge^k A$$

(*I meant closed in the standard topology on the $\text{Hom}$-space. I am not sure these questions are equivalent, since I am not very familiar with algebraic geometry. However, I am interested in the closedness question on its own.)

Here is a more concrete phrasing:

Let $b_{(i_1,\dots,i_k),(j_1,\dots,j_k)} \in \mathbb{R}$, where $(i_1,\dots,i_k),(j_1,\dots,j_k)$ are increasing multi-indices of order $k$ ( $1 \le i_1 <i_2 < \dots<i_k\le d$), under what conditions there exist a $d \times d$ matrix $A$, such that $ b_{(i_1,\dots,i_k),(j_1,\dots,j_k)}$ is the $k$-minor of a $A$, corresponding to rows $(i_1,\dots,i_k)$, and to columns $(j_1,\dots,j_k)$.