Let A and B be any two 3x3 matrices. If A is symmetric and B is skew-symmetric, then the matrix AB - BA is:

(AB-BA)T = (AB)T - (BA)T
= BTA^T - A^TBT
= -B^TA^T - A^TB^T (Since B is skew-symmetric, BT = -B and since A is symmetric A^T = A)
= -(BA) - (AB)
= -(BA + AB)
= - (AB + BA)
Also, (AB+BA)^T = (AB)^T + (BA)^T = B^TA^T + A^TB^T = -BA + AB
If the matrix AB - BA is skew-symmetric, then (AB - BA)^T = -(AB - BA)
-(AB+BA) = -AB + BA, which is not true.
Therefore, AB - BA is not necessarily skew-symmetric.
If AB - BA is symmetric, then (AB - BA)^T = AB - BA
-(AB + BA) = AB - BA
-AB - BA = AB - BA
-2AB = 2BA which is not true in general.
Therefore, AB - BA is not necessarily symmetric.
Thus the matrix AB - BA is neither symmetric nor skew-symmetric.