What makes the skin an organ rather than tissue?

The skin is classified as a organ because there are multiple cell types working together to function as a whole. The skin has several types of epithelial cells, including cells to make the skin, cells to make hair and cells that secrete sweat. In addition, there are fibroblasts and adipocytes in the dermis and hypodermis.

A tissue is a collection of a single type of cell working together to function as a whole; a reasonable (although not perfect) example of this would be a tendon or ligament.