Characterization of the subspaces of $\mathbb R^{m\times n}$ induced by rank-1 matrices?

I assume by "simple" you mean computationally efficient. It is unlikely that efficient algorithm for determining the existence of such a basis.

The sparse basis case is basically equivalent to the "matrix sparsification" problem. It is NP-hard to even approximate the sparsest basis for a linear space (see https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.144.450&rep=rep1&type=pdf). So under the widely believed assumption that P$\neq$NP, there is no efficient algorithm for the case of sparse matrices.

The problem of finding a low-rank basis can easily be seen as generalizing the sparse basis problem, by embed the vector space into the space of diagonal matrices. A sparse basis corrsponds to a basis of sparse diagonal matrices, which has low rank. Therefore, the low-rank case is also NP-hard.

As far as I know, the case where you are promised that there is a rank-1 basis has not been proven hard, but has also not been proven easy. This specific problem was mentioned here: https://link.springer.com/content/pdf/10.1007/s10107-016-1042-2.pdf. It seems very unlikely that the rank-1 case would turn out to be easy, given that the problem in general is hard.