Database Design - multiple tables
I hesitate to recommend this answer to a self described novice, but it seems to fit the case pretty well.

If were working this problem, I would go with a star schema approach. There would be three dimension tables:  Employees, Skills, and Levels.  And there would be one fact table, with foreign keys to all three tables, and no other columns (aka fields).  

This particular fact table only conveys a count, and no other measure.  

I might go ahead and make the hierarchy of skills a snowflake off of the skills dimension.  

If you are new at database design none of this would be familiar to you at all.  When people begin to work with star schemas,  they usually have a few years of database design under their belt.  But just in case you have access to a good tutorial on star schema,  you might want to follow through here.

In addition, if I could fit it into the budget, I would want to buy some sort of business intelligence software that would allow the managers could use to browse this data without learning the table structure.  Failing in that,  I would just try to build in the kind of drill down capabilities the managers need into my app. 

I hope all of this isn't mystifying.  Don't worry if it is.  Basically the fact table is capturing a many to many to many relationship,  which probably isn't covered in your learning materials.