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No. It is illegal to give or sell your Social Security number. Not because you have a Social Security number it means it belongs to you, a Social Security number is property of the government of the United States of America.
If you need to change it for a very special reason you need to go to government agencies to do that.
I do not think you can give it up even if you became a citizen of another country, the number will always be linked to you until you die. There is no other way for the USA government to track you down otherwise.
"The number is issued to an individual by the Social Security Administration, an independent agency of the United States government."
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