This is Eczema. The more you scratch the worse it itches.
Eczema is a skin disease that causes itchy, red, dry flaky skin. There is no cure or reasonable treatment. The treatment is typically steroids but there are side effects.
You should cut down on the amount of baths you get. Don't use hot water. Use a oil based soap like oilatum or Dove all natural. Use natural laundry detergents based on enzymes. Don't use bleach or dryer pads. Use a good moisturizer just after shower. You need to moisturize at least twice a day.
For itching oatmeal or salt baths work. Deep sea salt works well.
There are a lot of supplements that you can try. But no evidence that they work. Don't waste money on allergy tests. But you definitely have allergies. Food is usually the culprit.
Eggs, wheat, peanuts, soy, spicy food, milk and to much meat are typical culprits. A diet of fruit and vegetables can help a lot.
Definitely stay away from all processed and fast food.
Do Not Scratch! Apply pressure or soak in cool tub.