The moduli stack $Bun_{SL_n}X$ of $SL_n$-principal bundles over a projective curve $X$ is a $\mathbb{G}_m$-torsor over the sublocus of $Bun_{GL_n}X$ corresponding to vector bundles with zero determinant.

Is there a similar description for $Bun_{Sp_{2n}}X$? Edit: by this I mean a description as a torsor over something (e.g. a substack of $Bun_{GL_n}X$ for $SL_n$, which is not the same as the descriptions of its objects as vector spaces + something)

Also, by a theorem of Grothendieck the map of stacks $Bun_{O_n}X\to Bun_{GL_n}X$ induces an injection at the level of moduli spaces, and I think this essentially means that $Bun_{O_n}$ is a relative gerbe over its image (is this correct?). Is it possible to describe $Bun_{O_n}$ as a torsor over something (maybe a substack of $Bun_{GL_n}X$, as for $SL_n$?)