I think this is a well knowing result but I can't find any reference,
Let $(E,q)$ be a vector bundle with a non degenerated quadratic form $q:E\rightarrow E^*$ with trivial determinant, suppose that $q$ is symmetric (resp. skew-symmetric).
My question: Is there an equivalent of the stack of such couples with the one of $(\mathbb Z/2, SL_r)$-bundle, with the action of $\mathbb Z/2$ on $SL_r$ is giving by $$g\rightarrow ^tg^{-1} \;(resp.\,\,g\rightarrow J_n\;^tg^{-1}J_n)$$
where $J_n=\begin{pmatrix}0&I_n\\-I_n&0\end{pmatrix}$
and how can one prove that (if the answer was yes!)?
thanks.