This is a consequence of Maclean's theorem.  Since the $n$-torus itself is flat, a basis of the harmonic $1$-forms on $T^n$ are linearly independent at every point, so it follows from the description that Maclean gives in his theorem that the small deformations of $T^n$ as a special Lagrangian torus must be an $n$-parameter family that foliates a neighborhood of $T^n$ in $X$.