  We give a positive answer to the Aleksandrov problem in $n$-normed spaces
under the surjectivity assumption. Namely, we show that every surjective
mapping preserving $n$-distance one is affine, and thus is an $n$-isometry.
This is the first time to solve the Aleksandrov problem in $n$-normed spaces
with only surjective assumption even in the usual case $n=2$. Finally, when the
target space is $n$-strictly convex, we prove that every mapping preserving two
$n$-distances with an integer ratio is an affine $n$-isometry.
