prove that the diagonal elements of a scew symmetric matrix are all zero

We know that for a skew symmetric matrix :-

aij = -aji.

Consider a 3x3 matrix

according to the condition :-

Since , aij= -aji

a11 = -a11

implies a11+a11 = 0 ;

2a11 =0 , hence a11 = 0;

Similarly,

a12 =0 and a13 =0.

Hope it helps :)