Let $G(k,V)$ be the Grassmannian of $k$-planes in a complex vector space $V$ of dimension $n$. There is the famous universal exact sequence of vector bundles on $G(k,V)$ $$ 0 \to S \to V \otimes \mathcal O \to Q \to 0. $$ Moreover, one can define the tangent bundle $T_{G(k,V)}=\operatorname{Hom}(S,Q)$.
Explicitly, what can we say about $\operatorname{Hom}(S,Q^\vee)$?