The Riemann tensor is a function of the metric tensor when the Levi-Civita connection is used.
I'm wondering if there is a way to have an explicit form of the converse, so that having the coefficients of the Riemann tensor, can I have back the metric tensor supposing the Levi-Civita connection is used?
It depends on the precise formulation of the question. As written, the answer is negative. For instance, the torus $T^2$ has continuum of pairwise non-isometric metrics with zero curvature (zero Riemann tensor).
On the other hand, there are also positive results (such as Cartan's theorem) in the direction of recovering metric from curvature, see for instance section 8.2 in do Carmo's book "Riemannian Geometry." See also this MO question and references there.