we can derive it through the defination of antisymmetric. if (x,y) and (y,x) exist x=y.
so take a set of (1,2,3) now find all relations. of a*a
total 9 ,diagonal elements like (1,1)(2.2)(3,3) should be included.
total number of choices for them is 2^n ( i.e. n elemets have 2 choices either has to come or do not appear.)
total number of choices for for lower diagonal elements will be 3.
because either the lower diagonal should come or the upper diagonal elements or non of them comes.
3^(n2-n/2) three choices for these elemnts