  We give an alternative lower bound for the numerical radii of Hilbert space
operators. As a by-product, we find conditions such that
  \begin{equation*} \omega\left(\left[\begin{array}{cc} 0 & R \\ S & 0
\end{array}\right]\right)=\frac{\Vert R \Vert +\Vert S\Vert }{2}
\end{equation*} where $R, S \in \mathbb{B}(\mathcal{H})$.
