e_{123}  = 0 if any of i, j, k are equal = 1 if i,j, k are unequal and in cyclic order = 1 if i,j, k are unequal and not in cyclic order 
This can be defined by a determinant
e_{123}= 

This represents the cross product:
c = a × b
or in tensor notation:
c_{i} = e_{ijk} a _{j} b _{k}
then expanding out gives:
c_{1} = e_{123} a _{2} b _{3} + e_{132} a _{3} b _{2} = a _{2} b _{3}  a _{3} b _{2}
c_{2} = e_{231} a _{3} b _{1} + e_{213} a _{1} b _{3} = a _{3} b _{1}  a _{1} b _{3}
c_{3} = e_{312} a _{1} b _{2} + e_{321} a _{2} b _{1} = a _{1} b _{2}  a _{2} b _{1}