It is not necessary to bathe your skink all the time.
Sometimes they will walk through their poo and that creates a stinky mess or during difficult sheds a bath can help but in your case, the tiny dust particles from the substrate are not going to go away unless you chang to a different type of substrate so your skink will likely always be a bit dusty looking (: Not a huge deal.
If you feel comfortable putting your reptile in your own personal bathtub (not my cup of tea because many times they go to the bathroom in the water

) I would make certain there are no soap residues or cleansers left in the tub beforehand.
I think it is easier and more sanitary to just buy a plastic tub and keep it for only this purpose.
Water temp should be room temp, it can seem fine to us but may be too hot for the skink and would be stressful. Better to be safe. Also if the water is too high they may panic, it could also get into their ears or they could breathe it into their lungs so please, not too deep. An inch or two should be fine!