We have a tensor associated with a magnetic field in special relativity defined as $B_{jk} = \partial _j A_k - \partial _k A_j$ and two related equations: $B^i = \frac {1}{2} \epsilon ^{ijk} B_{jk}$ $\iff$ $B_{jk}= \epsilon _{jkl} B^{l}$, where $A$ is a vector field and $\epsilon$ is a Levi-Civita symbol. Even though I have a subtle knowledge about tensor algebra, but nonconsistent, I don't understand why is in the second equation that $1/2$. After I plugged the third equation into the second I got:
$B^i = \frac {1}{2} \epsilon ^{ijk}\epsilon _{jkl} B^{l} = \frac {1}{2}B^{l}\epsilon ^{i}_{l} $ ?