First off, you should know before anything else that no one here is going to do the work for you. It is Daniweb policy that we assist you in solving your problem, but that we not solve the problem directly by providing code. It doesn't sound like that is what youu are after, but it still needs to be said.
Second, I don't know if anyone here is going to be familiar enough with the specific area of expertise needed. I hope there is, but I won't promise it. Most of the people here are software engineers rather than mathematicians or physicists.
Third, the question is ambiguous, as there is no single Levi-Civita tensor. I am no mathematician, but from what is said on the subject on Wikipedia, there are invariant, co-variant and contra-variant tensor fields of this type, and while thelatter two at least can be inter-converted, they are different enough that you would need to know which type of Levi-Civita tensor field you are working with.
My best advice is to start by developing a general matrix class, which could be used as the starting point for the actual tensor field data structure. Once you can abstract away the matrix, you can handle it in a way suitable for applying the appropriate functions to the determinant of the matrix.