Soap can absolutely cause a rash! When a rash develops on the skin due to contact with an irritating chemical, we call this contact dermatitis. Contact dermatitis rashes tend to be reddened and itchy, occasionally with scaling or with small blisters. New laundry detergents are one of the most common causes of contact dermatitis.
I would recommend that you discontinue using this laundry detergent entirely to see if things get better. Of course, you will need to re-wash all the clothing that you have already washed with this detergent in a different detergent in order to remove any remaining traces. You may also want to try applying some over the counter anti itch cream, such as hydrocortisone cream, to the areas of rash to help them clear up more quickly and to keep you from scratching, which could predispose you to infection.
If the rash does not clear up with discontinuing the soap and using hydrocortisone, you should see your primary care doctor
. Sometimes contact dermatitis can be severe enough that it requires prescription medications to clear up. Alternatively, this rash could be from a different cause altogether, and your primary care doctor can help you figure that out.