Symmetric power of Euler sequence

Question

Is there 'good' representation for the sheaf $\ker(S^2V^* \otimes\mathcal{O}_{\mathbb{P}(V)} \twoheadrightarrow \mathcal{O}_{\mathbb{P}(V)}(2))$?

Answer 1

This sheaf $K$ fits into the following exact sequence $$ 0 \to S^2(\Omega(1)) \to K \to \Omega(2) \to 0. $$