In the following we demonstrate how it can be done with the new tensor capabilities of _Mathematica 9_.  
Let's define `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}}}

Now we define a tensor product of `v` and `LeviCivitaTensor[3]`:

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


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

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

In order to demonstrate even a bit a simpler example of tensor contraction let's 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` or `Tr` or simply computing the sum of diagonal elements:

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