Proof That Any Skew Symmetric Thrid Order System Has at Most One Non Zero Distinct Component

Letbe a completely skew symmetric third order systemwithcomponents.

Consider any component with at least two identical indices egInterchanging any two components changes the sign so that

The only possible non zero components are those with no indices being equal

The only possible components are a)123 , a)132, a_213, a_231, a_312 , a_321 .

But sinceis skew symmetricand there is at most one non zero component.