A symmetric matrix is always a square matrix. If A be any square matrix and if A = AT, then matrix A is called symmetric matrix.

Let A = [ aij ] be a square matrix and if A = AT, it means if we interchange rows and columns of A, then the matrix remains unchanged i.e. aij = aji, for all i and j. So, we say that, in a symmetric matrix, all elements are symmetric with respect to main diagonal.
A = $\begin{bmatrix}
2 &-1 &0 \\
-1 &-2 &3 \\
0 &3 &4
\end{bmatrix}$ AT = $\begin{bmatrix}
2 &-1 &0 \\
-1 &-2 &3 \\
0 &3 &4
\end{bmatrix}$

Here A =AT. So, A is a symmetric matrix. All diagonal matrices are symmetric because in diagonal matrix all non-diagonal elements are zero.

A square matrix say B = [ bij ] is said to skew-symmetric, if and only if B = -BT
i.e. [ bij ] = - [ bji ].

Let A = $\begin{bmatrix}
0&a &b \\
-a &0 &-c \\
-b &c &0
\end{bmatrix}$

Now AT = $\begin{bmatrix}
0&-a &-b \\
a &0 &c \\
b &-c &0
\end{bmatrix}$

so A = AT

For symmetric and skew-symmetric matrices, the necessary condition is that the matrices should be square matrices. And for skew-symmetric matrix, all the diagonal elements are zero.