Only elementary objects can be saved out. The SVG (or indeed also PDF) standard only contains basic strokes, everything else is a layer that your editor provides and is expanded on save. The standard only defines paths, strokes and dashes. No art brushes, no pattern brushes, no variable width strokes etc.
So no you can not do this on brushes. Same applies to PDF though not necessarily EPS (although in practice also there) its only embedding illustrator specific info that allows illustrator to reverse the action.
However you can release the brush assignment by setting default appearance on the shape, or use the corresponding freehand path tool that does not by default attach a brush.