You will want to use the 'log' operation, usually bound to Ctrl-a H. This opens a new file called screenlog.N where N is the window number. If the file exists, then output is appended to the file. All output in that window is logged to the screenlog.N file until you deactivate by pressing Ctrl-a H again. You can also get this functionality with Ctrl-a :log. You can change the name using the logfile {filename} command. The file gets placed in the directory in which screen was first started, but that can be changed with the logfile command or with chdir [{directory}].
Note that this is not the same as hardcopy or the scroll buffer. This is the same functionality as the script(1) program, but tied to a single window and without calling a new program.
If you want to save the entire contents of the scroll buffer, use Ctrl-a ] g <space> G <space> Ctrl-a :writebuf This saves the scroll buffer to the file /tmp/screen-exchange. if you want to save only a portion, then change the 'g' and 'G' to other movement keys.