There are many subtleties to this but it sounds like you haven’t included the Bianchi identities, in particular the first and perhaps the second Bianchi identities. There are reasons why this tensor has these properties. There are also the anti-skew symmetries and one of them is to do with defining a suitable inner product on the tangent space T(M) which is induced by the metric tensor g_ab.
If a curvature tensor satisfies the skew symmetry, anti skew symmetry and first Bianchi identities then it is trivial to show that there exists a curvature tensor at each point p in M (a pseudo-Riemmannian manifold) which has n^2(n^2-1)/12 components.
It’s all in two standard texts:
-Introducing Einstein’s Relativity by D’Inverno
-The Mathematical theory of Black Holes by Chandrasekhar.
