I have this equation
$ \varepsilon_{ijk}B_k = \partial_iA_j - \partial_jA_i $
and I was asked to prove $\mathbf{B}=\nabla\times\mathbf{A}$, where $\mathbf{B}=B^i\mathbf{e}_i$ and $\mathbf{A}=A^i\mathbf{e}_i$, $i=1,2,3$.
Multiplying this equation by $\varepsilon^{ijk}$ and using that $\varepsilon_{ijk}\varepsilon^{ijk}=6$ doesn't work and I end up with an additional multiplicative factor of $\frac{1}{3}$. On the other hand, if I multiply the equation by $\varepsilon^{ijl}$ and use $\varepsilon^{ijl}\varepsilon_{ijk}=2\delta^l_k$ instead, I get the right answer.
What is the mistake with my first approach?
You can't multiply by $\epsilon^{ijk}$ because there's already a $k$ in $B_k$ and indices should not appear more than twice.