Recommended approach when we deal with **symbolic tensors** should use the new and powerful  [capabilities][1] of _Mathematica 9_. We can exploit appropriate tensor functionality also when we have defined a tensor `v`. e.g.:

    v = RandomInteger[{-10, 10}, {3, 3, 3}]
>     {{{8, 4, -4}, {2, -2, 4}, {-1, 9, -8}},
     {{-9, -7, 8}, {-9, -4, 5}, {-9, 2, -7}}, 
     {{-7, 10, -4}, {-5, 9, -9}, {-4, 6, 5}}}

we sholud create a tensor product of `v` and `LeviCivitaTensor[3]`:

    tp = TensorProduct[ v, LeviCivitaTensor[3]];

Now we have to contract the tensor product `tp`  having `6` indicies in their appropriate pairs, namely
`{1, 4}, {2, 5}` and `{3, 6}`:

    TensorContract[ tp, {{1, 4}, {2, 5}, {3, 6}}]
>     -7

Tensor contraction of a tensor product can be performed also  using `Inner` or in an obvious (more procedural) way, using  `Sum`, `Part` and `Times` :

    Sum[ v[[i, j, k]] LeviCivitaTensor[3][[i, j, k]], {i, 3}, {j, 3}, {k, 3}]
>     -7

As a bonus we demonstrate even a bit a simpler example of tensor contraction, i.e. we find a trace of a matrix `m`:
 

    m = RandomInteger[{-10, 10}, {3, 3}]
>     {{ 9, 3, -8}, 
     {-2, 4, -6}, 
     { 9, 1, -7}}

We can see that it should be `6` using `TensorContract` (contracting with respect to the first and the second indicies , i.e. using `{ 1, 2}` as the second argument in `TensorContract` or `Tr` or simply computing the sum of diagonal elements:

    TensorContract[ m, {1, 2}]
    Tr @ m == TensorContract[ m, {1, 2}] 
>      6
     True 


  [1]: http://reference.wolfram.com/mathematica/guide/SymbolicTensors.html