The module of sections of the cotangent bundle for complex projective space is finitely generated projective as the cotangent bundle is locally trivial. Thus the module of sections has a dual basis. But what is the dual basis explicitly? Say, on the coordinate chart where $(u_1,\dotsc,u_n)$ is sent to the equivalence class of $(u_1,\dotsc,u_n,1)$ what is the formula for the dual basis? This problem is of course solvable, but the answer must be known and in a reasonably simple form. I would be very grateful for someone sending me in the right direction.