Since S is symmetric, it can be diagonalized by an unitary matrix. But the decomposition I am looking for is to decompose S into orthogonal matrices. In this case, s is not guaranteed to be diagonal but it has to be symmetric. SchurDecomposition gives a $O^{\dagger}$ instead of $O^T$ on the RHS, and the JordanDecomposition will just give the diagonal matrix.
I searched that s is called the symmetric normal form, even with this name I cannot find how to actually perform the task. Many thanks for given hints!
Your first sentence "Since S is symmetric, it can be diagonalized by an unitary matrix", is false over $M_n(\mathbb{C})$.
Its true only when $S$ is normal, that is, $SS^*=S^*S$.
Moreover, any $n\times n$ complex matrix is similar to a symmetric matrix; in particular, a symmetric complex matrix may be non-diagonalizable...