Hello purakanui,

the Levi-Civita-Symbol is generally a tensor:

mapping the n-fold cartesian product of a vector space to its underlying field (usually the real or complex numbers).

However if you represented it as matrix, it'd have to have only 2 indices. Of course one can define the Levi-Civita-Symbol with 2 indices:

else.

Then a matrix representing this would look like this:

However, I have to warn you: This E is actuallyreally a matrix.NOT

At least it doesn't behave like a matrix under the usual multiplication of a matrix with vectors.

So better leave the tensor notation of the Levi-Civita-symbol as it is instead of writing it as an array.

If you want to express the Levi-Civita-symbol in another basis, then you have to apply basis transformations on it,

, using the summation convention, where

are now really matrices,

e.g.

But use the summation convention to calculate the values!

I have to remark that it might well be that the Levi-Civita-Symbol is defined as it is, ineverybasis, but I don't know.

Here is how the Levi-Civita-Symbol with 3 indices may be represented:

Visualization of the Levi-Civita symbol as a 3×... - Visual Hints - Quora