You could try:
transform[a_] := Map[Flatten] @* Transpose @* Map[p |-> Partition[p, Dimensions[a][[1]]]]@a
Though, for larger matrices it wouldn't perform well. For starters, the composite @* operator is not too fast nor does invoking Dimensions for each row contribute to a quick solution. These concerns can easily be addressed by rewriting it as a module:
transform2[a_] := Module[{r = Dimensions[a][[1]]},
Map[Flatten, Transpose[Map[Partition[#, r]&, a]]]]
