tannakian description of vector bundles Let $S$ be a scheme and $E$ be a rank $n$ vector bundle on $S$. $E$ corresponds to a $GL_n$ torsor $P$ via the definition 
$$
P = Isom_S(E,\mathbf{A}_S^n)
$$
Tannakian theory tells us that $P$ corresponds to a fiber functor
$$
\eta : Rep(GL_n) \to Vect_S.
$$
(Recall that $\eta(V)$ is defined to be the pushout of $P$ via the representation $GL_n \to GL(V))$
My questions is : is there an easy description of $\eta$ in terms of $E$ without going through $P$ ?
 A: Write $BGL_n$ for the classifying stack of rank $n$ vector bundles; in principle it is not necessary to know what a $GL_n$ torsor is in order to say this. $E$ is represented by a map $f : S \to BGL_n$, and pullback along this map is a functor
$$f^{\ast} : QC(BGL_n) \to QC(S).$$
This is $\eta$, or rather it is a "large" version of $\eta$: you can identify $QC(BGL_n)$ with representations of $GL_n$. (If you like, this is the only non-tautologous statement I'm making: you can define $QC(BGL_n)$ to consist of natural assignments of quasicoherent sheaves on $S$ to rank $n$ vector bundles on $S$.) To pass to vector bundles, $f^{\ast}$ is symmetric monoidal, so induces a corresponding functor on dualizable objects. 
A: How is your vector bundle $E$ presented? (As a locally free sheaf? As it's total space? Or...) If you like to think in terms of transition functions, i.e., Čech cocycle $\underline{\sigma} = \{\sigma_{ij}\} \in H^1(S,GL_n)$ (of non-abelian cohomology group), then the representation $(\pi,V) \in Rep_k(GL_n)$ will be sent to the vector bundle corresponding to the Čech coycle $\{ \pi(\sigma_{ij}) \} \in H^1(S,GL_V)$.
To get an idea of what is going on in the previous answers: This is essentially "resolving" $S$ using a cover $\mathfrak{U}=\{U\}$ and the  Čech nerve $C^\bullet(\mathfrak{U})$ and thinking of the stacks $BGL_n$ and $BGL_V$ as simplicial schemes via the Milnor bar constrcution. We get that the bundle $\eta(\pi,V)$ is classified by the composite
$C^\bullet(\mathfrak{U}) \to BGL_n^\bullet \to BGL_V^\bullet$
where the first morphism (is the classifying map and) is precisely the data $\underline{\sigma}$ and the second is $B\pi$ (the functor $B$ applied to $\pi : GL_n \to GL_V$).
Is at least the first description closer to what you wanted?
