From any short exact sequence $0\longrightarrow F\longrightarrow G\longrightarrow H\longrightarrow 0$, we can construct the following induced exact sequence on wedge product (following apendix A):
$$ 0\longrightarrow \bigwedge\nolimits^2 F\longrightarrow \bigwedge\nolimits^2 G\longrightarrow (F\otimes H)\oplus\bigwedge\nolimits^2 H\longrightarrow 0 $$
I'm now trying yo apply this to the Euler short exact sequence:
$$ 0\longrightarrow\mathcal{O}_{\mathbb{P}^n}\longrightarrow\mathcal{O}_{\mathbb{P}^n}(+1)^{\oplus n+1}\longrightarrow\mathcal{T}_{\mathbb{P}^n}\longrightarrow 0 $$
In that case we would have:
$$ 0\longrightarrow\bigwedge\nolimits^2\mathcal{O}_{\mathbb{P}^n}\longrightarrow\bigwedge\nolimits^2\left(\mathcal{O}_{\mathbb{P}^n}(+1)^{\oplus n+1}\right)\longrightarrow\left(\bigwedge\nolimits^2\mathcal{T}_{\mathbb{P}^n}\right)\oplus\mathcal{T}_{\mathbb{P}^n}\longrightarrow 0 $$
My question is if this last short exact sequence makes sense, in particular, if $\bigwedge\nolimits^2\mathcal{O}_{\mathbb{P}^n}$ really makes sense.