Question: Given an invertible linear map $U:V\to V$, consider the induced map $\tilde{U}:\Lambda^k(V)\to \Lambda^k(V)$ given by $$\tilde{U}(v_1\wedge \cdots\wedge v_k):=\sum_{j=1}^kv_1\wedge \cdots \wedge Uv_j\wedge \cdots\wedge v_k.$$ Is there a nice formula for $\tilde{U}^{-1}$ in terms of $U^{-1}$?
Context: in quantum field theory, such a formula would allow one to solve for a many-electron Green's function in terms of a single-electron Green's function.