This question is related to this old question.
Let $X$ be a scheme over a field $k$ and $G=GL_n$. A tensor exact functor $$ \omega: Rep_k(GL_n)\to Bun_X $$ is equivalent to a $GL_n$-torsor (as a sheaf) by taking $$ P:=\underline{Isom}(\omega_{can},\omega) $$ and furthermore this yields a vector bundle $$ V_1:=P\times^{Gl_n} \Bbb{G}_a^n.$$ On the other hand I can take $$ V_2:= \omega(k^n,id).$$
Q: I think that these two approaches are equivalent, i.e. $V_1\cong V_2$. Is that correct and if so why?
My idea was to prove that equivalently there is an isomorphism of $GL_n$-torsors $$ P\to \underline{Isom}(\mathcal{O}_X^n,V_2)$$ but I failed to make this isomorphism concrete.