Can my detergent give my family eczema?

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.
Not technically. Eczema, or atopic dermatitis, is a hereditary inflammatory condition. It is associated with food & airborne allergies but there is no definite cause. Irritants such as fragrance in detergents can worsen itch & increase eczema, so many physicians recommend double rinsing clothes before wearing to wash away all detergents. Therefore, detergents can exacerbate eczema but won't directly cause it.