Let \( M \) denote the set of all real matrices of order \( 3 \times 3 \) and let \( S=\{-3,-2,-1,1,2\} \). Let \( S_{1}=\{A=[a_{ij}] \in M: A=A^{T} \text{ and } a_{ij}\in S, \\ \forall \ i,j\} \), \( S_{2}=\{A=[a_{ij}] \in M: A-A^{T} \text{ and } a_{ij}\in S, \\ \forall \ i,j\} \), \( S_{3}=\{A=[a_{ij}] \in M: a_{11}+a_{22}+a_{33}=0 \text{ and } a_{ij}\in S, \\ \forall \ i,j\} \). If \( n(S_{1}\cup S_{2}\cup S_{3}) = 125\,\alpha \), then \( \alpha \) equals ______.
