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.