Psoriasis: This is a common skin condition characterized by thick, scaly, red plaques often found on the elbows, knees, scalp, and lower back. The distribution of the rash on the back, neck, and elbows is typical for psoriasis, making it the most likely diagnosis.

Atopic Dermatitis (Eczema): Although more common in children, atopic dermatitis can occur in adults and may present with a rash on the neck, elbows, and back. It's often associated with itching and a personal or family history of atopy.
Seborrheic Dermatitis: This condition causes a rash that can appear on the scalp, face, and sometimes on the chest and back. While less typical for the elbows, it's a common skin condition that could be considered.
Contact Dermatitis: An allergic reaction or irritation from substances like soaps, detergents, or metals can cause a rash. Areas exposed to these substances, such as the neck (from jewelry) or elbows (from certain fabrics), could be affected.

Lyme Disease: Caused by a tick bite, Lyme disease can initially present with a distinctive "bull's-eye" rash (erythema migrans) but can also cause a more generalized rash. Given its potential for serious complications if untreated, it's crucial not to miss this diagnosis.
Syphilis: The secondary stage of syphilis can cause a rash on the palms of the hands and soles of the feet, but it can also appear on other parts of the body, including the back, neck, and elbows. Syphilis is a significant diagnosis to consider due to its potential for severe health consequences if left untreated.
Meningococcemia: This is a severe infection caused by Neisseria meningitidis. While the rash associated with meningococcemia is typically petechial (small spots due to bleeding under the skin), it's a life-threatening condition that requires immediate medical attention.

Pityriasis Rubra Pilaris (PRP): A rare skin disorder characterized by constant scaling, redness, and thickening of the skin. It can start with a reddish-orange rash on the face, then spreads to other parts of the body, including the elbows and back.
Keratosis Pilaris: A condition that causes small, rough patches on the skin, often on the arms, thighs, cheeks, and buttocks, but can occasionally appear on the elbows and back. It's more of a cosmetic concern but can be considered in the differential diagnosis for a rash in these areas.