Definition of the (anti-)symmetric part of a tensor

Question

Consider the following tensor: $$ A_{ij} = B_{k,i}C_{k,j} $$ which can be decomposed into its symmetric and antisymmetric parts: $A_{ij} = A_{ij}^s + A_{ij}^a$. I'm all confused as to how to express these parts: if I use transpose of the tensor, I get \begin{align} A_{ij}^s = \tfrac{1}{2} (A_{ij} + A_{ji}) = \tfrac{1}{2}(B_{k,i}C_{k,j} + C_{j,k}B_{i,k}) \\ A_{ij}^a = \tfrac{1}{2} (A_{ij} - A_{ji}) = \tfrac{1}{2}(B_{k,i}C_{k,j} - C_{j,k}B_{i,k}) \end{align} but they are not symmetric/antisymmetric upon interchange of $i,j$. The more sensible decomposition would be using (anti-)symmetrization \begin{align} A_{ij}^s = B_{k,(i|}C_{k,|j)} = \tfrac{1}{2}(B_{k,i}C_{k,j} + B_{k,j}C_{k,i}) \\ A_{ij}^a = B_{k,[i|}C_{k,|j]} = \tfrac{1}{2}(B_{k,i}C_{k,j} - B_{k,j}C_{k,i}) \end{align} which does the job, but it's different from the standard transposition method.