$p^{n+1}+p^{n}-p$ = count 2x2 $F_p$-matrices satisfying Grassman algebra relations - true? (And the same for symmetric 3x3 Grassmans)

Question

It seems number of 2x2 matrices over finite field $F_p$ satisfying Grassman algebra relations ( $\psi_i ^2=0, \psi_i \psi_j + \psi_j \psi_i = 0 , i=1...n$) is equal to $p^{n+1}+p^{n}-p$. (Guessed by simulation).

Argument and/or a reference would be welcome.

Question 2: Exactly the same count appears to be for symmetric 3x3 matrices satisfying the same Grassman algebra relations - true ?

PS

While for antysymmetric there seems to be no solution except trivial one even for the constraint $\psi_i^2 =0$.