SO cord is a heavy rubber cord with fine stranded wire. It is designed to be rough-housed around.
I would not recommend using a ROMEX (the white house-style wire) for an extension cord because the wire is solid and will break much more easily. I'm not saying it wouldn't work but you would really be wasting your money unless you are intending to use the ROMEX wire inside your shed for a more permanent installation later. Remember that you must consider the distance from the house to the shed when sizing the wire. A #8 wire might be sufficient for a distance of 20' but you might have to bump up to a #6 wire size for 50' or #4 up to 75'. Again, I don't have a wire-load sizing chart in front of me, but you get the idea.
Another idea you might consider is getting long welding leads. You could plug the welder in at the house and just run the leads where you need them. I am not sure what the price per foot of 2/0 copper welding lead versus 3 conductor #8 copper SO cord.
