  Given a graph $G$, let $G^\sigma$ be an oriented graph of $G$ with the
orientation $\sigma$ and skew-adjacency matrix $S(G^\sigma)$. Then the spectrum
of $S(G^\sigma)$ consisting of all the eigenvalues of $S(G^\sigma)$ is called
the skew-spectrum of $G^\sigma$, denoted by $Sp(G^\sigma)$. The skew energy of
the oriented graph $G^\sigma$, denoted by $\mathcal{E}_S(G^\sigma)$, is defined
as the sum of the norms of all the eigenvalues of $S(G^\sigma)$. In this paper,
we give orientations of the Kronecker product $H\otimes G$ and the strong
product $H\ast G$ of $H$ and $G$ where $H$ is a bipartite graph and $G$ is an
arbitrary graph. Then we determine the skew-spectra of the resultant oriented
graphs. As applications, we construct new families of oriented graphs with
maximum skew energy. Moreover, we consider the skew energy of the orientation
of the lexicographic product $H[G]$ of a bipartite graph $H$ and a graph $G$.
