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I have a matrix in element form as: $$A_{ij}= \sum_{m,n=1}^3\epsilon_{jmn} B_{im} C_{jn}$$ where $B$ & $C$ are also $3\times 3$ matrices. How should I use Levi-Civita symbol in Mathematica so that the element of $A$ matrix can be found?

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I think you can just use it as expected...

SeedRandom[42]; (* Just so the example is reproducible *)
ε=LeviCivitaTensor[3]
b=RandomInteger[100,{3,3}]
c=RandomInteger[100,{3,3}]
a=Table[Sum[ε[[j,m,n]] b[[i,m]] c[[j,n]],{m,1,3},{n,1,3}],{i,1,3},{j,1,3}]

(5418 -5204 -792

279 -1404 372

-1104 -8254 2644 )

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