SetDirectory["~/KappaLib/"];
<<kappaLib-1.2.m

Section 4.3: Compute matrix 

   epsilon = (epsilon^IJ)_IJ

Table[emOrd[i],{i,1,6}]//MatrixForm

mat=Table[
Signature[
{(emOrd[i])[[1]],(emOrd[i])[[2]],(emOrd[j])[[1]],(emOrd[j])[[2]]}],
{i,1,6},
{j,1,6}
];
mat//MatrixForm

Extra: epsilon coincides with 6x6 matrix representation of the Euclidean Hodge star operator

g=DiagonalMatrix[{1,1,1,1}];
kappa=emHodge[g];
emKappaToMatrix[kappa]//MatrixForm