Aside: I'm not signed up for the site (not interested in tying that login to other accounts), but I wanted to comment that luisb's explanation of Eq. (13) seems

Aside: I'm not signed up for the site (not interested in tying that login to other accounts), but I wanted to comment that luisb's explanation of Eq. (13) seems to miss the point a little. That equation doesn't need to be derived from Eq. (4). Instead, Eq. (13) is just a statement that any antisymmetric 3x3 matrix can always be recharacterized as a (pseudo)vector "times" the totally antisymmetric Levi-Civita symbol epsilon_{ijk}. As the text says, Eq. (13) is the definition of H. Only after we've defined what H is can we then use Eq. (4) to provide a definition of E.