So I understand that a matrix A = sym(A) + skew(A) = 1/2(A + Atranspose) + 1/2(A - Atranspose)

I cant seem to understand why multiplying any matrix A by a symmetric or skew matrix B is equal to B.sym(A) or B.skew(A), respectively. in other words:

B.A = B.sym(A) when B is symmetric and

B.A = B.skew(A) when B is a skew matrix

any help is appreciated. (B.A is the inner product of B and A in case there is any confusion)