A Drazin invertible operator T ∈ B(H) is called skew D-quasi-normal operator if T * and TT D commute or equivalently TT D is normal. In this paper, firstly we give a list of conditions on an operator T, each of which is equivalent to T being skew D-quasi-normal. Furthermore, we obtain the matrix representation of these operators. We also develop some basic properties of such operators. Secondly we extend the Kaplansky theorem and the Fuglede-Putnam commutativity theorem for normal operators to skew D-quasi-normal matrices,