flying from taipei to shanghai and back to taipei via macau

Hello!

I want to go to Hangzhou for a weekend. This I would like to do by flying to Shanghai, then take the train to Hangzhou (Cheaper) and then afterwards I need to get back to Taipei. I was wondering if it really is not possible to fly from Taipei to Shanghai and back (I have a Dutch passport, so I will be allowed to get a 144-hour vis). And if it wouldn't be possible to just fly back and forth with this visa, would it be allowed to fly from taipei to shanghai and then back flying from shanghai to macau and from there back to taipei?

Thank you very much for an answer!