Is there an obvious way to write 
$T\_{\mathbb{P}^2}\otimes T\_{\mathbb{P}^2}$ and $S^2(T\_{\mathbb{P}^2})$?



Symmetric power of tangent space
One can write explicit resolutions by line bundles ---
$$
0 \to O \to O(1)^6 \to O(2)^9 \to T \otimes T \to 0
$$
and
$$
0 \to O(1)^3 \to O(2)^6 \to S^2T \to 0
$$
respectively.


