In electrodynamics this vector potential A taken together with the electrical scalar potential ρ give the four-vector Aμ=(ρ,A)T, and the Maxwell tensor can then be written as Fμν=dμAν-dνAμ. Remembering that FijijkBk, you can recover the equation that Blush Response has provided above:
    εijlFij = εijlεijkBk = (δjjδkljkδjl)Bk = 2Bi
    εijlFij = εijl(djAl-dlAj) = 2εijldjAl
Which is just B=curl A.
The fact that the Maxwell tensor can be written in terms of the vector potential implies half of Maxwell's equations - when expressed in the form dμFνρ+ dρFμν+ dνFρμ=0. Quite remarkable really.