Outdoor outlet to outlet wiring
There's no problem with using stranded wire, as long as it's properly sized. The problem with your plan, is that you want to use an extension cord.  

An extension cord is meant to be used as a temporary wiring method, with the cord intact and undamaged. Once you cut the end off the cord, the cord is no longer UL listed, and cannot be used in a code compliant manner.

You can use any cable or wire that is listed for wet/outdoor locations, whether it be solid or stranded wires.  Underground Feeder (type UF) cable is a common choice, if you want to use cable.  If you choose to use cable, you'll have to make sure that it's protected from physical damage.  

THWN wire would be acceptable, if you want to run it through conduit.