  We study $n$-dimensional matrices with $\{0,1\}$-entries ($n$-cubes) such
that all their $2$-dimensional slices are incidence matrices of symmetric
designs. A known construction of these objects obtained from difference sets is
generalized so that the resulting $n$-cubes may have inequivalent slices. For
suitable parameters, they can be transformed into $n$-dimensional Hadamard
matrices with this property. In contrast, previously known constructions of
$n$-dimensional designs all give examples with equivalent slices.
