If $A$ is a $10\times10$ matrix with entries from the set $\{0, 1, 2, 3\}$ and if $AA^T$ is of the form: $$\begin{pmatrix} 0 & * & * & \cdots & * \\ * & 0 & * & \cdots & * \\ * & * & 0 & \cdots & * \\ \vdots & \vdots & \vdots & \cdots & \vdots \\ * & * & * & \cdots & 0 \end{pmatrix}$$
Then the number of such matrices $A$ is:
A) $(4^3)^{10}$
B)$(4^2)^{10}$
C)$4^{10}$
D) $1$
Since all the diagonal elements of $AA^T$ is zero I could realize that it is a skew-symmetric matrix. But, I'm no able to understand how I can use this result for finding the possibilities of the original matrix. I would like hints rather than answers.