1) I was wondering if there exists a criterion (of (a) combinatorial or (b) geometric nature) for a sum of simple vectors $V\in\wedge^k(\mathbb R^n)$,

$V=e_{a_{11}}\wedge\cdots\wedge e_{a_{1k}} + \cdots + e_{a_{m1}}\wedge\cdots\wedge e_{a_{mk}}$, where $e_1,\cdots, e_n$ is a basis of $\mathbb R^n$

to be a simple vector: in other words, I want to know what should one verify in order to be sure that $V$ can be written as $V=v_1\wedge\cdots\wedge v_k$ for some vectors $v_i\in\mathbb R^n$.

2) What happens if we allow noninteger coefficients? In other words, how does the answer change if we consider question 1) for

$V=c_1e_{a_{11}}\wedge\cdots\wedge e_{a_{1k}} + \cdots + c_me_{a_{m1}}\wedge\cdots\wedge e_{a_{mk}}$, where now $c_i\in\mathbb R$.

3) any reference or hint about some (name of) branch of mathematics dealing with this is also welcome!