As I understand so far, the metric tensor of a Riemannian manifold is an $n \times n$ matrix in many specific examples. As such it could formally be the curl of some vector potential or just the derivative.
I wonder if this is indeed possible and if yes, if it is interesting or really just a formal coincidence.
If I do not misunderstand your question your are asking if it is possible to define a more primitive object than the metric tensor, on a smooth manifold $M$, from which we can derive the metric tensor $g$, isn't it?. Such a more primitive object of course must be called potential. This approach recall me the so called Kaehler manifolds https://en.wikipedia.org/wiki/K%C3%A4hler_manifold
Kaehler manifolds form a subclass of Riemannian manifolds. For them there is indeed a concept of potential. Namely, around each point of a Kaehler manifold the metric tensor can be recovered from a function called potential. The recovering procedure is something like taking the Hessian of the potential.
The following post develops the idea of Hessian type metric on a Riemannian manifold: https://mathoverflow.net/questions/122308/when-a-riemannian-manifold-is-of-hessian-typ