Your mom was right. She knew her stuff when she told you to wear gloves when you wash dishes. I get it. Gloves can be annoying and kinda make us look like a 50s housewife, but trust me when I say it's totally worth it.
If you wash dishes with ungloved hands, you are also washing off the natural oils, (aka protection). Harsh soaps and hot water can cause major issues in time with dry skin, rashes, dermatitis, and irritation. And keep in mind that a lot of us have been using tons of hand sanitizer due to the plague and all, which is harsh and very drying for the skin.