Can matrices do cross product with vectors? this is a practical problem raised from my chemistry research, apologize if somewhere I used incorrect words.
I'm dealing with the derivatives of a vector with respect to another vector. After some searching, I found the rule for differentiating a column vector with respect to a row vector is (all these vectors are 3 dimensional in Cartesian):
$$
\mathbf{e}=\begin{bmatrix}e_1\\e_2\\e_3\end{bmatrix}
$$
$$
\mathbf{P}^T=\begin{bmatrix}P_1 & P_2 & P_3 \end{bmatrix}
$$
$$
\frac{\partial \mathbf{e}}{\partial \mathbf{P}^T}=
\begin{bmatrix} 
\frac{\partial e_1}{\partial P_1} & \frac{\partial e_1}{\partial P_2} & \frac{\partial e_1}{\partial P_3} \\
\frac{\partial e_2}{\partial P_1} & \frac{\partial e_2}{\partial P_2} & \frac{\partial e_2}{\partial P_3} \\
\frac{\partial e_3}{\partial P_1} & \frac{\partial e_3}{\partial P_2} & \frac{\partial e_3}{\partial P_3} \\
\end{bmatrix}
$$
Here my $\mathbf{e}$ is determined from a cross product, that is, $\mathbf{e}=\mathbf{n}\times\mathbf{m}$. So I tried to do this:
$$
\frac{\partial \mathbf{n}\times \mathbf{m}}{\partial \mathbf{P}^T}
$$
From wiki, it says the product rule can be applied, so I arrived here:
$$
\frac{\partial \mathbf{n}\times \mathbf{m}}{\partial \mathbf{P}^T}=\frac{\partial \mathbf{n}}{\partial \mathbf{P}^T}\times\mathbf{m}+\mathbf{n}\times\frac{\partial \mathbf{m}}{\partial \mathbf{P}^T}
$$
Now I'm confused because $\frac{\partial \mathbf{n}}{\partial \mathbf{P}^T}$ and $\frac{\partial \mathbf{m}}{\partial \mathbf{P}^T}$ are matrices, and they do cross product with vectors. 
I didn't found anywhere saying the cross product of matrix, so I'm concerned perhaps there is something wrong in the derivation.
Hope you can give some help. Thanks in advance!

The writing is a bit awkward. If you take the derivative with respect to $P_i$ then 
$$ \frac{\partial  n\times m}{\partial P_i} = \frac{\partial  n}{\partial P_i}\times m + n\times \frac{\partial  m}{\partial P_i} $$
makes more sense. If you want to give a meaning to $\frac{\partial  n}{\partial P}\times m$ then it should be that the cross product acts column wise, i.e. that you take each column vector in $\frac{\partial  n}{\partial P}$ and replace it by its cross product with m (which is equivalent to the above formula).
