What is the nature of the term 什么都?
This might help you to understand it better.
我已经(把)什么都学会了.
And 我已经什么都学会了 just omits 把.
Another example: 我把这本书读完了. Colloquially, you can say 我这本书读完了.
More: 我把话说完了/我话说完了。
But the omission doesn't always work well.  E. g.  我把这件事给忘了.  It doesn't sound right to say 我这件事给忘了.  In this case, we do need 把 and can't be omitted.
So be cautious of this omission in practice.