Let $X$ be a scheme over an algebraically closed field (I'm mainly interested in the characteristic zero case), $V$ a rank $n$ vector bundle and $\ell$ a line bundle on $X$, and
$\phi:V\otimes V\to\ell$
an $\ell$-valued quadratic form on $V$, which can be seen as a map $\phi:V\to V^{*}\otimes\ell$. The form $\phi$ is generically non degenerate if it induces an isomorphism
$\det(\phi):\det (V)\otimes\mathcal{O}(D)\to\det(V^{*})\otimes\ell^{\;n}$
for some divisor $D$ on $X$.
My question is: is it possible for a form $\phi$ to be generically non degenerate yet not to induce an isomorphism $V\otimes\mathcal{O}(D)\to V^{*}\otimes\ell$ for some divisor $D$?