Curvature and connection in gauge theory

Question

My question is about gauge theory in minkowsky or euclidean spacetime, with a non-abelian lie group $G$.

Is the connection $A_\mu^a$ determined (modulo gauge redundancy) by the curvature $G^a_{\mu\nu}$($=\partial_\mu A^a_\nu-\partial_\nu A^a_\mu+2gf^a\,_{bc}A^b_\mu A^c_\nu$)? Or putting this in another way, is the curvature inyective modulo gauge simmetry?

Thanks.