To update a Tutor's Teaching Skills, go to People > Tutors > select the Tutor and click the Add button on the 'Teaching Skills' panel. This will open the 'Update Skills' page, from where you can choose a new Subject and skill level (the level of attainment that Tutor has) from the respective dropdown menus. Here you can also add more Teaching Skills, or remove them if needed.
Adding a new Teaching Skill to a Tutor's profile.
Note: Any Hidden Subjects will not appear in the Subject dropdown. If a Subject you expect is missing, check System > Settings > People and Activity > Hidden Subjects.