And technically yes
If the family did not have eczema
before and you have recently developed the eczematoid rash
, then there is a possibility for a detergent to cause this.
Some detergents contain stain fighting enzymes as additives. These are active proteins that can cause itching as they breakdown your body oils. By definition eczema is: an itch that rashes. So, if it makes you itch, it can give you eczema.