So, bit of a weird question here, but let's just say someone has a clan tattoo on their cheek and it was cut off, would the flap of skin last if it was just then sealed in a box? If so, then how long for?
Bare skin is not going to last long. First of all it will lose humidity and dry. This will lead to shrinking and stiffening/embrittlement of the skin. Then the bacteria naturally living on the skin would start to decompose it.
If you want to preserve skin you need to tan it:
Tanning hide into leather involves a process which permanently alters the protein structure of skin, making it more durable and less susceptible to decomposition, and also possibly coloring it. Before tanning, the skins are unhaired, degreased, desalted and soaked in water over a period of 6 hours to 2 days. Historically this process was considered a noxious or "odoriferous trade" and relegated to the outskirts of town.
Another alternative is to make rawhide
The skin from buffalo, deer, elk or cattle from which most rawhide originates is prepared by removing all fur, meat and fat. The hide is then usually stretched over a frame before being dried
If it is in a box, @L.Dutch covers it in his answer.
But it is freaky, this keeping of skin. Maybe you are ok with a little freakier. You could keep the skin alive by allografting it onto someone.
Pictures of skin grafts are a little too horrific for the stack I think. But these mice are cute! Mouse skin swap from http://rstb.royalsocietypublishing.org/content/370/1666/20140382
Skin grafts are done all the time. Usually they find a swath of good skin somewhere else on the patient, harvest the top half of it and move it to where it is needed. People who need a lot of skin (burn patients) can use frozen cadaver skin. Skin from live donors can be used too. Usually these grafts are rejected but not always. From: http://www.woundsresearch.com/article/prolonged-allograft-survival-patient-chronic-immunosuppression-case-report-and-systematic
In an immunosuppressive state, it is possible that the skin allograft will not be rejected as long as the other transplanted organs are not being rejected as well. Without rejection, the skin allograft will be able to vascularize and provide long-term wound coverage, as well as protect from further infectious agents, making this a viable option to treat similar cases.
So you could keep your skin good on some host either with a 1: very suppressed immune system or 2: genetically identical to the owner of the skin; his twin.
I can imagine a creepy story where some individual is the repository for many tattoos removed from various people.
Just for the sake of clarity, a skin flap is a region of tissue that is still attached to the body (ie. has a blood supply) and advanced over a deficient area.
It all depends on humidity, temperature, and contamination level of tissue. (Was the skin scrubbed beforehand? Was it removed with sterile equipment? Did it get popped into the refrigerator right away? Cooled quickly or slowly? Frozen? How thick is it? Just the epidermis/dermis or was the whole cheek w/ muscle removed?)
If the tissue is completely detached, it will not have a blood supply and be starved of all the vital nutrients it needs. Commensal skin bacteria start to grow and if there is contamination those bacteria (maybe even fungi) start to grow as well. There are so so many factors that are not your friend here when it comes to preserving the tissue/proper wound healing. If it's to be preserved like mentioned above, you'd have to pin it on each side to something firm before preserving it because the tissue is going to shrink and good luck preserving the tattoo.