The answer is yes, many people do just that, but you can get in trouble if you don’t run your pump runs 24 hours a day. We would not recommend putting tablets in your skimmer if you operate your pool on a timer and you can also get in trouble if the power goes off.
The chlorine tablets are concentrated and they have a very low ph. When your pump is running, the water going through your skimmer is constantly eroding the tablet, adding chlorine to the water which quickly makes its way through your filter system and into your pool.
If the pump goes off, now that tablet is sitting in just a few gallons of water and the chlorine level rises and ph drops. Now you have a skimmer full of highly chlorinated/acidic water and when it sloshes out of the skimmer and back into the pool with the regular movement of the water in your pool, that caustic water runs right down the face of the wall and can burn your liner as shown in the photos below. This will damage and permanently discolor your liner.