Let $V$ be a $k$-dimensional vector space, and consider a decomposable tensor in $\bigwedge^\ell V \otimes \bigwedge^m V$ having the form $$ \mathbf{v} \otimes \mathbf{w} := v_1 \wedge \cdots \wedge v_\ell \otimes w_1 \wedge \cdots \wedge w_m, $$ where $\ell,m \leq k$. Now, if $\ell = m = k$, then we have "exchange relations'' (i.e., the Plucker relations, i.e., Sylvester's lemma), as follows: for any choice of fixed $w_{i_1}, \ldots, w_{i_p}$, with $p \leq m$, we have $$ \mathbf{v} \otimes \mathbf{w} = \sum_E E(\mathbf{v}\otimes \mathbf{w}), $$ where the sum ranges over all exchanges $E$ which swap all of the $w_{i_j}$'s with distinct vectors $v_i$ (preserving the order of the $w_{i_j}$'s). In fact, this still holds true if we let $m \leq k$.
In the case where both $\ell,m < k$, the exchange relation above no longer holds. Nonetheless, I have observed some similar relations. For example, again choosing vectors $w_{i_1}, \ldots, w_{i_p}$ where $k-\ell < p \leq m$, we have $$ \sum_E (-1)^{|E|} E(\mathbf{v} \otimes \mathbf{w}) = 0, $$ where the sum ranges over all exchanges $E$ of all possible subsets of $\\{w_{i_1}, \ldots, w_{i_p}\\}$, and $|E|$ denotes the number of $w_{i_j}$'s which $E$ exchanges. This is not too hard to prove, but seems far "weaker" than the previous exchange relation, since it contains many more terms. Surely there are "sparser" relations as well?
Are there other well-known relations in this setting? Or is there a standard reference that addresses these sorts of questions?