tee is a command which displays or pipes the output of a command and copies it into a file or a variable.

The -a option appends the output to the end of file instead of writing over it.

You can also create an alias in ~/.bashrc to record your session when using ssh:

function sshlog () { \ssh $@ 2>&1 | tee -a $(date +%Y%m%d).log; }
alias ssh=sshlog