(make -k all 2>&1) | tee error_log
Will push everthing to the error_log as well as scrolling it up the screen ( so you know whats going on. )
If you simply want to record the errors, and you don't want them echoed to the screen then.
make -k all 2> error_log
The only thing you seemed to be missing above was the space after >
If you want to still echo the errors, and record them to file do:
(make -k all 3>&1 1>&2 2>&3 ) | tee error_log