What a terrific question and although it seems logical to be able to do so, I would not. I am not a doctor, perhaps one of them will be able to confirm for me, though from what I understand adult drugs can have side effects in children that can make the illness worse.
The primary example I found of this is when using medications that are targeting the digestive system, such as anti-diarrheal medications like Pepto Bismol that contain salicylate (leading to Reye's Syndrome) and loperamide (that may prolong the sickness by slowing down the digestive tract and keeping the bug in your child's system longer).
I recommend talking to your pediatrician before offering your child any form of medication - especially adult forms.
In health and happiness!