Am I liable if someone runs over my skateboard and it does damage?

Whose fault it is depends on the precise facts. You write that "a little later" someeone ran over the skateboard. If that was more than, say, a few tens of seconds later, then you would certainly be at fault, since you would have had an opportunity to retrieve the skateboard (i.e. if you did anything else, or paused/delayed before getting the skateboard, you would be at fault).

As for the driver, whether he/she was at fault depends on whether it was "reasonable" that he/she should have seen the skateboard. A skateboard is a low-visibility item--it is easy for a driver to miss. If the driver was otherwise driving carefully, it is quite possible that a court would conclude that there was nothing unreasonable about not seeing a skateboard in the street--or seeing it but not having time to react. Only if the driver was in some way driving carelessly at the time (e.g. talking on a cell phone or texting; driving too fast; eating in car or applying make-up; etc.) would it likely be found to be that the driver was at fault, too.

Again, the issue is not whether the driver "could" have seen the skateboard--it's whether the average reasonable driver, driving reasonably carefully, would have, and would have done so in sufficient time or distance to avoid it.