Every time i finish washing dishes or washing my hands and my arm gets wet i get this rash looking thing on my arm that burns a little bit. why?

Raynaud's: First i would consider an allergic reaction to the detergent. Other possibility is raynaud's phenomenon in which there is an exaggerated blood vessel constriction when exposed to cold (in this case while your hands are wet). I suggest you check with your physician for proper diagnosis and treatment. Hope you feel well soon.

Rash after washing : If the skin of the arms and hands is dry, it can be easily irritated by dish detergents, soaps and hot water. Moisturizing regularly should help improve this, as will using rubber gloves with cotton gloves inside them for doing housework. If only rubber gloves are used, hands will sweat inside them and still be irritated. Other products that irritate hands include sanitizing gels.

Not sure: Please see your dermatologist for evaluation and to discuss treatment options. Could be from hot water or from chemicals in the water.