$\displaystyle \displaystyle{\epsilon_{ijk}=\begin{cases}

0, &\text{if any index is equal to any other index}\\

+1, &\text{if}\;i, j, k\;\text{form an {\it even} permutation of}\;1,2,3\\

-1, &\text{if}\;i, j, k\;\text{form an {\it odd} permutation of}\;1,2,3

\end{cases}.}$

An even permutation has an even number of exchanges of position of two symbols. Cyclic permutations (for example, $\displaystyle 123\to 231\to 312$) are always even. Thus

$\displaystyle \epsilon_{122}=\epsilon_{313}=\epsilon_{211}=0,\qu ad\text{etc.}$

$\displaystyle \epsilon_{123}=\epsilon_{231}=\epsilon_{312}=+1$

$\displaystyle \epsilon_{132}=\epsilon_{213}=\epsilon_{321}=-1$