I am trying to find an explicit way to view global holomorphic sections of $\Omega^{1} \otimes \mathcal{O} (2)$ over $\mathbb{CP}^{2}$. I guess what I mean by "explicit" would be a formulation over an affine open $U_i \subset \mathbb{CP}^{2}$. According to what I found in Okoneck, Schneider and Spindler, there is a 3dimensional space of such sections, but I want this for a computation in differential geometry.
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If $x,y,z$ are coordinates on $P^2$ then the 3 sections of $\Omega(2)$ are given by $xdy  ydx$, $ydzzdy$, and $zdxxdz$. 


Use the Euler sequence: $0 \to \Omega^1_{\mathbb{P}^n_A/A} \to \mathcal{O}_{\mathbb{P}^n_A}(1)^{\oplus n+1} \to \mathcal{O}_{\mathbb{P}^n_A} \to 0.$ Everything can be seen explicitly from here. Tensoring the sequence with $\mathcal{O}(2)$ gives an exact sequence, which is still exact if you take global sections because every monomial of degree 2 is a multiple of a monomial of degree 1.:) So the dim you want = 96 =3. 

