Let \begin{align*} A&= \begin{bmatrix} 1 & -1\\1 & 1 \end{bmatrix} \end{align*}
Then \begin{align*} \begin{bmatrix} 1 & -1\\1 & 1 \end{bmatrix} \begin{bmatrix} 1 & 1\\-1 & 1 \end{bmatrix} &= \begin{bmatrix} 2 & 0\\0 & 2 \end{bmatrix} = \begin{bmatrix} 1 & 1\\-1 & 1 \end{bmatrix} \begin{bmatrix} 1 & -1\\1 & 1 \end{bmatrix} \end{align*} so we see by Definition NRML that $A$ is normal. However, $A$ is not symmetric (hence, as a real matrix, not Hermitian), not unitary, and not skew-symmetric.