Let $X$ be a smooth general ordinary Gushel-Mukai threefold. There is an embedding $X\rightarrow\mathrm{Gr}(2,5):=G$. Consider the normal bundle $\mathcal{N}_{X|G}$, how to compute cohomology of this vector bundle and related vector bundle, i.e, what is $H^k(X,\mathcal{N}_{X|G}\otimes\mathcal{O}_X(2))$, $H^k(X,\mathcal{N}_{X|G})?$

Let $T_X$ be the tangent bundle of $X$($X$ is general OGM threefold or more general, Fano threefold of index 1, Picard rank 1), how to compute $H^0(X,\bigwedge^2T_X)$? I was trying to use normal bundle exact sequence and some other exact sequence involving tangent bundle on $X$, but I could not decide which one I should use?