Let $V$ be a vector space of dimension $>n$, and define the subset $$ K:=\{ ([\omega],v)\mid v\wedge\omega=0 \}\subset\mathbb{P}(\Lambda^nV)\times V\, . $$ Denote also by $\pi:K\longrightarrow \mathbb{P}(\Lambda^nV)$ the restriction of the canonical projection onto the first factor of $\mathbb{P}(\Lambda^nV)\times V$. Then, $\pi$ is a linear bundle over $\mathbb{P}(\Lambda^nV)$ with variable fibre dimension.

QUESTION 1: is it true that $$\{[\omega]\in \mathbb{P}(\Lambda^nV)\mid \dim\pi^{-1}([\omega])=n\}\quad\quad\quad (^*)$$ is precisely the Plucker image of the Grassmanniann $\mathrm{G}(n,V)$?

In the affirmative case, I'd like to know whether such a "bundle" $\pi$ ever appeared, perhaps in other guises, and ask the next

QUESTION 2: what if I replace $n$ with an arbitrary $i\neq n$ in the above $(^*)$? What do I obtain?

I'm expecting some subset of $\mathbb{P}(\Lambda^nV)$ intrinsically related to $\mathrm{G}(n,V)$, like its tangent/secant variety.

(I'm aware of another MO question, concerning the Plucker embedding and the tautological bundle, and another one, concerning its ring of functions, though neither can help here.)