Is there a bijective correspondence between metric tensors and coordinate systems?

Question

Let's suppose I'm in a flat space described by cartesian coordinates, and that I define some new curvilinear coordinates by leaning on my cartesian coordinates. It is easy, at least in principle, to find metric tensor. I wonder if there is a bijective correspondence between coordinates systems and metric tensors, and if there is a way to jump from metric tensors to coordinates systems that generate them.