The Levi-Civita symbol, which is not a tensor, is defined as

\begin{align}

{\widetilde{\epsilon }}_{{\mu }_1{\mu }_2\dots {\mu }_n}=\left\{ \begin{array}{ll}

+1 & \mathrm{if\ }{\mu }_1{\mu }_2\dots {\mu }_n\mathrm{\ is\ an\ even\ permitation\ of}\ 01..(n-1)\ \\

-1 & \mathrm{if\ }{\mu }_1{\mu }_2\dots {\mu }_n\mathrm{\ is\ an\ odd\ permitation\ of}\ 01..\left(n-1\right) \\

0 & \mathrm{otherwise} \end{array}

\right. & \phantom {10000}(1) \\

\end{align}Instead of ##\widetilde{\epsilon }## it is also often written as ##\widehat{\varepsilon }##, which is what I usually do.

We usually only need it in three or four dimensions so it becomes much simpler:

\begin{align}

{\widehat{\varepsilon }}_{\alpha \beta \gamma \delta }=\left\{ \begin{array}{ll}

+1 & \mathrm{if\ }\alpha \beta \gamma \delta \mathrm{\ is\ an\ even\ permitation\ of}\ 0123\ \\

-1 & \mathrm{if\ }\alpha \beta \gamma \delta \mathrm{\ is\ an\ odd\ permitation\ of}\ 0123 \\

0 & \mathrm{otherwise} \end{array}

\right. & \phantom {10000}(2) \\

\end{align}or

\begin{align}

{\widehat{\varepsilon }}_{ijk}=\left\{ \begin{array}{ll}

+1 & \mathrm{if\ }\alpha \beta \gamma \delta \mathrm{\ is\ an\ even\ permitation\ of}\ 123\ \\

-1 & \mathrm{if\ }\alpha \beta \gamma \delta \mathrm{\ is\ an\ odd\ permitation\ of}\ 123 \\

0 & \mathrm{otherwise} \end{array}

\right. & \phantom {10000}(3) \\

\end{align}One simple thing is that if any two of the indices on ##\widehat{\varepsilon }## are the same, it vanishes.

Even in four dimensions it is fairly easy to calculate, for example

\begin{align}

{\widehat{\varepsilon }}_{1023}=-{\widehat{\varepsilon }}_{0123}=-1 & \phantom {10000}(4) \\

\end{align}It's easy to jot down the indices and determine the sign by walking indices in one direction or another. Here is an example from my real sad life

\begin{align}

{\widetilde{\epsilon }}_{{\mu }_1{\mu }_2\dots {\mu }_n}=\left\{ \begin{array}{ll}

+1 & \mathrm{if\ }{\mu }_1{\mu }_2\dots {\mu }_n\mathrm{\ is\ an\ even\ permitation\ of}\ 01..(n-1)\ \\

-1 & \mathrm{if\ }{\mu }_1{\mu }_2\dots {\mu }_n\mathrm{\ is\ an\ odd\ permitation\ of}\ 01..\left(n-1\right) \\

0 & \mathrm{otherwise} \end{array}

\right. & \phantom {10000}(1) \\

\end{align}Instead of ##\widetilde{\epsilon }## it is also often written as ##\widehat{\varepsilon }##, which is what I usually do.

We usually only need it in three or four dimensions so it becomes much simpler:

\begin{align}

{\widehat{\varepsilon }}_{\alpha \beta \gamma \delta }=\left\{ \begin{array}{ll}

+1 & \mathrm{if\ }\alpha \beta \gamma \delta \mathrm{\ is\ an\ even\ permitation\ of}\ 0123\ \\

-1 & \mathrm{if\ }\alpha \beta \gamma \delta \mathrm{\ is\ an\ odd\ permitation\ of}\ 0123 \\

0 & \mathrm{otherwise} \end{array}

\right. & \phantom {10000}(2) \\

\end{align}or

\begin{align}

{\widehat{\varepsilon }}_{ijk}=\left\{ \begin{array}{ll}

+1 & \mathrm{if\ }\alpha \beta \gamma \delta \mathrm{\ is\ an\ even\ permitation\ of}\ 123\ \\

-1 & \mathrm{if\ }\alpha \beta \gamma \delta \mathrm{\ is\ an\ odd\ permitation\ of}\ 123 \\

0 & \mathrm{otherwise} \end{array}

\right. & \phantom {10000}(3) \\

\end{align}One simple thing is that if any two of the indices on ##\widehat{\varepsilon }## are the same, it vanishes.

Even in four dimensions it is fairly easy to calculate, for example

\begin{align}

{\widehat{\varepsilon }}_{1023}=-{\widehat{\varepsilon }}_{0123}=-1 & \phantom {10000}(4) \\

\end{align}It's easy to jot down the indices and determine the sign by walking indices in one direction or another. Here is an example from my real sad life