By that definition, skin is definitely an organ. Skin is made up of very specific cells and tissues, and their collective purpose is to act as the boundary between "you" and "the world." One of the interesting things about skin that makes it different from the internal organs is that it has to interact extensively with the real world. Therefore, it is loaded with sensors, and it also has a very tough, layered design so it can handle diverse environmental factors, such as abrasion and sunlight.
If you take a look at the cross-section of typical skin (like the skin on your arm or leg) you find that it is made up of two main layers: the epidermis on the outside, and the dermis on the inside. The epidermis is the barrier between you and the world, while the dermis is the layer containing all the "equipment" -- such as nerve endings, sweat glands, hair follicles, and so on. Here's a description to help you understand what is going on:
In the skin's subcutaneous layer you can see the blood vessels. These vessels branch extensively into the dermis to supply the sweat glands, hair follicles, sebaceous glands, and erector muscles with blood. They also fan out into the dermis' capillary bed.
