Could be done as below.
a = Array[x, {3, 3}];
aatrans[i_, j_] := Sum[a[[i, k]] a[[j, k]], {k, 3}]
Table[aatrans[i, j], {i, 3}, {j, 3}] // Expand
(* Out[635]= {{x[1, 1]^2 + x[1, 2]^2 + x[1, 3]^2,
x[1, 1] x[2, 1] + x[1, 2] x[2, 2] + x[1, 3] x[2, 3],
x[1, 1] x[3, 1] + x[1, 2] x[3, 2] +
x[1, 3] x[3, 3]}, {x[1, 1] x[2, 1] + x[1, 2] x[2, 2] +
x[1, 3] x[2, 3], x[2, 1]^2 + x[2, 2]^2 + x[2, 3]^2,
x[2, 1] x[3, 1] + x[2, 2] x[3, 2] +
x[2, 3] x[3, 3]}, {x[1, 1] x[3, 1] + x[1, 2] x[3, 2] +
x[1, 3] x[3, 3],
x[2, 1] x[3, 1] + x[2, 2] x[3, 2] + x[2, 3] x[3, 3],
x[3, 1]^2 + x[3, 2]^2 + x[3, 3]^2}} *)
But a.Transpose[a]
is the better way to do this in general.