Basically, you really only need to use your full name on legal documents, or on something like an application to a school or job. It'd sound weird to introduce yourself as "Alexander" to your peers unless you dislike the name shortened to "Alex", or to write the full name out on non-legal documents.
Though, my husband's middle name was "James" and on legal documents he would sign it as "Jm" which is apparantly quite acceptable to do with the name James, though on the line where he had to print his name he'd print it as "James". My husband's first name can also be shortened, and he'd also print the full first name out, but shorten his first name in the signature as well. I also had to look at some very old records on some microfilm, and on the censuses they would just write "Jm" instead of James, and often shorten Alexander to "Alex", and so on, but only if it was a middle name.