From the Euler sequence
$$0\to\mathcal{O}_{\mathbb{P}^n}\to V\otimes\mathcal{O}_{\mathbb{P}^n}(1)\to T_{\mathbb{P}^n}\to0$$
it is easy to deduce that $T_{\mathbb{P}^n|L}=\mathcal{O}_L(2)\oplus\mathcal{O}_L(1)^{\oplus(n-1)}$ for any line $L\subset\mathbb{P}^n$.
Now for any line $L\subset\mathbb{P}^n$ I want to compute $(\Lambda^2T_{\mathbb{P}^n})_{|L}$. For convenience suppose that $L=\{z_2=z_3=...=z_n=0\}$.
It seems true that after taking the second exterior power of the Euler sequence we obtain
$$0\to T_{\mathbb{P}^n}\to \Lambda^2V\otimes\mathcal{O}_{\mathbb{P}^n}(2)\to \Lambda^2T_{\mathbb{P}^n}\to0,$$ thus after the restriction to $L$ we obtain the exact sequence $$0\to\mathcal{O}_L(2)\oplus\mathcal{O}_L(1)^{\oplus(n-1)}\to \Lambda^2V\otimes\mathcal{O}_{L}(2)\to(\Lambda^2T_{\mathbb{P}^n})_{|L}\to0.$$
The problem is that I don't understand much what is the first map in this exact sequence so I can't figure out what is the cokernel.