I'm quite sure this should be classically known, however I am not an expert on the topic and I was unable to find a precise reference in the huge literature concerning Veronese embeddings and Grassmannians. Any pointer to relevant books or papers will be highly appreciated.

Let $V$ be a finite-dimensional vector space (over $\mathbb{C}$, say) and let $v_n \colon \mathbb{P}(V) \longrightarrow \mathbb{P}(S^n V)$ be the usual $n$th Veronese embedding.

If $\ell$ is a line in $\mathbb{P}(V)$, then $v_n(\ell)$ is a rational normal curve of degree $n$, that will be contained in precisely one $n$-plane $\Pi_{\ell} \subset \mathbb{P}(S^n V)$. Then we can define a morphism of projective Grassmannians $$\psi_n \colon \mathbb{G}(1, \, \mathbb{P}(V)) \longrightarrow \mathbb{G}(n, \, \mathbb{P}(S^nV)),$$ given by $\psi_n(\ell) :=\Pi_{\ell}$.

Question.Is $\psi_n$ an embedding?