How to Find the Extrinsic Geometry from the Intrinsic Metric Tensor?

Question

Working in 3D euclidian space, is it possible to find the parametrization of a surface in such space given only the intrinsic metric tensor of this surface? For example, supose this metric tensor was given: $$g_{uv}=\begin{bmatrix}1&0\\0&sin^2\left(u\right)\end{bmatrix}$$ And you wish to find out what the surface with such metric tensor looks in 3D euclidian space. We know the metric tensor of flat 3D space, then given this information, can one find the parametrization of this surface? In this case, this metric is the metric of a sphere, but supose it was another random metric, how to procede then? Given the parametrization it's very easy to find the metric, but how to do the opposite?