It is well known that an $SL_n$-bundle $E$ on an algebraic curve $X$ is self dual (i.e $E\cong E^*$) iff it is an $SO_n$-bundle
However, I can't see why, because the isomorphism $E\cong E^*$ means that the transitions functions $g_{ij}$ verifies $$^tg^{-1}=a_igb_j$$ for some $a_i$ and $b_j$,
So, I want to see a proof.
Thanks.