Let P=ABT−BAT
To determine the nature of this matrix, we find PT and compare it with P.
PT=(ABT−BAT)T
PT=(ABT)T−(BAT)T
Using (XY)T=YTXT:
PT=(BT)TAT−(AT)TBT
Using (XT)T=X:
PT=BAT−ABT
PT=−(ABT−BAT)
PT=−P
Since PT=−P, the matrix (ABT−BAT) is always a skew-symmetric matrix.