Could dark matter be a higher dimensional object? Why?

Well dark matter is, in a layman terms just a concept used to identify missing mass in the universe. (extremely simplified and probably wrong but look that up). It could be but then if it is then we would still be able to percive the matter in some form. All we know from instruments is that they have mass and absorb light. We would need more info before any conclusions but it is 100% a possibility. But remember that when someone says "3-d" for example they are talking about a certain space ... technically matter exists on all planes (at least all the planes we know of in our local universe.) The laws of the cosmos are not yet proven to be constant and even if they are what of outside it?

When you take high school physics ask your teacher.