Let $X$ be a smooth projective curve, and $E$ a vector bundle on $X$ such that there exist a bilinear perfect symmetric form $$E\otimes E\rightarrow \mathcal O_X$$

When I see $E$ as a $GL_r$ principal bundle, the existence of such a form is equivalent to the existence of a reduction of the structure group to the linear orthogonal group $O_r$, so $E$ is now an $O_r$ bundle, so the vector bundle $E$ admits transition functions $\{f_{ij}\}$ such that $$f_{ij}=\,^tf_{ij}^{-1}$$

Question: How can one prove directly without passing to the theory of principal bundles that given a vector bundle $E$ with a symmetric perfect pairing as above, then this bundle admits a transition functions as above?

Many thanks.