For simplicity, assume everythings occur on a smooth projective variaty $X$.

Dual bundle of the given line bundle $\mathcal L$ is determined by $\mathcal L$ and $c_1(\mathcal L)$.

$\mathcal L^*= \mathcal L (-2c_1(\mathcal L)) $

My question is, is their any similar relation between a vector bundle of rank >1 and its dual? Can dual vector bundle be described as a combination of original bundle and its data(e.g chern classes)? Even for a rank 2 case, I have no idea about this problem.

If you have one, please give me some short proof or sketch. Good reference is also very preferable. I appreciate any help.