What could be causing me to have an allergic skin rash in my own swimming pool but not in others?

Can be infection. Pseudomonas aeruginosa is a common cause of folliculitis, which might be what you are having. It is commonly called "hot tub folliculitis" but can be spread in swimming pools, or even sometimes by shared diving suits. It is very important to keep tubs, pools, and shared water equipment clean on a regular basis. *sometimes is eczema, which is dry, sensitive skin, depends on type of rash.
Pool treatment. Some people are allergic to the shocking agent used for pools and hot tubs. Potassium peroxymonosulfate or ppms is an oxidizing agent used to keep pools clean. A study showed that allergies to this chemical aren't uncommon and tend to afflict men more than women. Treating the allergy with topical steroids and changing the pool's shocking agent can help.