If $T_i$ are the components of a covariant vector, show that $\left(\frac{\partial T_i}{\partial x^j}-\frac{\partial T_j}{\partial x^i}\right)$ are the components of a skew-symmetric tensor of rank 2.
My plan was to prove that $\left(\frac{\partial T_i}{\partial x^j}-\frac{\partial T_j}{\partial x^i}\right)$ are the components of a covariant tensor of rank 2 in step 1 and then showing that it is skew-symmetric in step 2.
So how to prove the above the statements of the two steps above?