(This question highly overlaps with this and also this.)

The irreducible ${\sf SL}_{n-1}$-module $V_{[1,1,\ldots,1]}$ is the one providing the minimal projective embedding $\mathbb{P}(V_{[1,1,\ldots,1]})$ for the complete flag variety $\mathrm{Fl}(\mathbb{C}^n)$.

However one does not need representation theory in order to realise that $\mathrm{Fl}(\mathbb{C}^n)$ is projective. First, embed $$ \mathrm{Fl}(\mathbb{C}^n)\subset\mathrm{Gr}(1,\mathbb{C}^n)\times\mathrm{Gr}(2,\mathbb{C}^n)\times\cdots\times\mathrm{Gr}(n-1,\mathbb{C}^n)\, , $$ then regard each Grassmannian $\mathrm{Gr}(i,\mathbb{C}^n)$ as a projective variety in $\mathbb{P}\bigwedge^i\mathbb{C}^n$, and finally use the Segre embedding: $$ \mathrm{Fl}(\mathbb{C}^n)\subset\mathbb{P}\left(\bigotimes_{i=0}^n\bigwedge^i\mathbb{C}^n \right)\, . $$

QUESTION. Is there any "evident map" defined on $\bigotimes_{i=0}^n\bigwedge^i\mathbb{C}^n$ whose kernel is $V_{[1,1,\ldots,1]}$?

By "evident" I mean definable in terms of elementary operations between tensors, like skew-symmetrisation. For instance, $V_{[1,1,\ldots,1]}$ lies in the common kernel of all the skew-symmetrisations $$ \bigotimes_{i=0}^n\bigwedge^i\mathbb{C}^n \stackrel{s_{ab}}{\longrightarrow} \bigwedge^{a+b}\mathbb{C}^n\otimes\bigotimes_{i\neq a,b}\bigwedge^i\mathbb{C}^n \, , $$ where $$ s_{ab}(\cdots\otimes\omega_a\otimes\cdots\otimes\omega_b\otimes\cdots):= (\omega_a\wedge\omega_b)\otimes\cdots\, , $$ and I suspect that $V_{[1,1,\ldots,1]}$ is exactly equal to $\bigcap_{a,b}\ker s_{ab}$, though I cannot prove it!

QUESTION (reformulated). Is there an "evident way" of regarding $V_{[1,1,\ldots,1]}$ as a submodule of $\bigotimes_{i=0}^n\bigwedge^i\mathbb{C}^n$?