Let $V$ be a vector subspace of $\mathscr{L}(E)$ containing $\operatorname{Id}_{E}$, included in $\operatorname{Sim}(E)$ and of dimension $d \geqslant 2$. Let $\left(\operatorname{Id}_{E}, f_{1}, \ldots, f_{d-1}\right)$ be a basis of $V$. Show that there exists a basis $\left(\operatorname{Id}_{E}, g_{1}, \ldots, g_{d-1}\right)$ of $V$ such that for all $i \in \{1, 2, \ldots, d-1\}$, $g_{i}$ is antisymmetric (one will seek $g_{i}$ as a combination of $f_{i}$ and $\operatorname{Id}_{E}$).