How to derive the connection from representation of structure group $Gl(n,\mathbb{C})$?

Question

Define the vector bundle in form of fiber product that is, $$ E=GL(E)\times_g\mathbb{C}^n $$ by $(pg,u)\sim (p,gu)$; And define the associated bundle just by a representation $T$ of $GL(n,\mathbb{C})$, that is $(pg,u)\sim (p,T(g)u)$. Given the local connection matrix of $E$ as $\Gamma$, what is the associated connection matrix $\Gamma'$ with representation $T$? For instance, we have $\Gamma'=-\Gamma^t$ if the associated bundle $E'=E^*$ the dual bundle. Thanks