I think all four versions are acceptable, though sentence 3 sounds slightly less natural than the others, at least to my ear.
It is acceptable to say "skin" because, in the given context, "skin" is an uncountable noun like "water" or "food". It refers to any amount of skin, large or small, or to the concept of "skin" as a whole.
It is also acceptable to say "the skin" because, in the given context, "the skin" is a body part. When we speak about a body part in general, we use the definite article. For example, we would say:
The heart pumps blood to the other parts of the body.
The knee is the largest joint in the body.
Here, "the heart" refers to a typical heart, not to the heart of any specific person. It is an "example" heart that stands in for all hearts. (See this other question for more details: Article with parts of the body)
Here are some example uses I found online:
Skin is the body's largest organ and, along with hair, nails, and skin-associated nerves and glands, is part of the integumentary system, according to the National Library of Medicine (NLM).
https://www.livescience.com/health/skin-facts-about-the-bodys-largest-organ-and-its-functions
The skin is the body’s largest organ, made of water, protein, fats and minerals.
https://my.clevelandclinic.org/health/articles/10978-skin
Skin is the human body's largest organ.
https://www.nationalgeographic.com/science/article/skin-1
To date, the skin is considered to be the largest organ.
https://www.healthline.com/health/largest-organs-in-the-body
What is the largest organ in the body? The skin is the largest of the more than 70 organs you possess.
https://www.usatoday.com/story/news/health/2022/09/24/largest-organ-in-the-body/10380205002/
Skin is the largest organ of the body.
https://www.americanskin.org/resource/
The skin, comprising roughly 16% of your body mass, is thought to be your largest organ.
https://www.cnn.com/2018/03/27/health/new-organ-interstitium-study/index.html
The skin that covers your body is called the dermal system. [...] Skin is the body's largest organ.
https://wwwn.cdc.gov/Tsp/substances/ToxOrganListing.aspx?toxid=2
As you can see, both forms are common.