Why is $Hom(T(-1),O(1)) \cong \Lambda^2(\mathbb{C}^{n+1})^*$ on $\mathbb{CP}^n$?

Question

I am currently trying to read a paper and the author claims the following. On $\mathbb{CP}^5$ we have $$Hom(T(-1),O(1)) \cong \Lambda^2(\mathbb{C}^{6})^*.$$ The proof is claimed to be a consequence of the Koszul complex. First of all, I believe by $Hom(T(-1),O(1))$, he actually means the space of sections of that bundle since otherwise the dimensions of the spaces don't match. I looked up the definition of the Koszul complex (in particular I saw Koszul complexes of twisting sheaves ) and have been playing around with the Euler sequence $0\rightarrow O(-1) \to \mathbb{C}^6 \to T(-1)\to 0$ and the wedge and exterior powers of these sequences. I am still clueless how to prove his statement and haven't been able to find a reference for it.