Using the elements –3, –2, –1, 0, 1, 2, 3

Trace is o is possible if principal diagonal is filled using  {(0,0,0),(1,-1,0)(2,-2,0)(3,-3,0)(1,1,-2)(-1,-1,2)(1,2,-3)(-1,-2,3) in 37 ways

and each of non diagonal elements can be filled in 7 ways

The number of $3\times 3$  matrices having trace 0 is $37\left(7^6\right)$

Using the elements –3, –2, –1, 0, 1, 2, 3

Total number of $3\times 3$ matrices=7⁶

$$\begin{gathered} \text{ For symmetric matrix, }{A}^{\mathrm{\top }}=A \\ Number of symmetric matrices=7^6 \end{gathered}$$

$$\begin{gathered} \text{ For skew-symmetric matrix, }{A}^{\mathrm{\top }}=-A \\ Number of skew symmetric matrices=7^3 \end{gathered}$$