Hello,
From your symptoms it can be dry skin or xersosis. Too much soapy water, exposure to harsh chemicals, the normal aging process and certain types of skin diseases are some of the causes of decreased amounts of protective skin oils which in turn causes dry skin.
The mainstay of management is liberal use of moisturizers after washing with luke warm water. Soap should be used minimally. Moisturizers should be reapplied liberally during the day. You can also use a topical cortisone (steroid) cream or ointment like dermacort(hydrocortisone). Adding a humidifier to the central heating system of your home or using a humidifier also helps.
If still the symptoms persist then possibility of eczema or atopic dermatitis, psoriasis, ichthyosis (fish scale-like skin) or pitted keratolysis is there. Please consult a dermatologist for clinical examination in that case as treatment is specific for each one.
It is very difficult to precisely confirm a diagnosis without examination and investigations and the answer is based on the medical information provided. For exact diagnosis, you are requested to consult your doctor. I sincerely hope that helps. Take care and please do keep me posted on how you are doing.