Passive air vent DIY questions

Kinda new DIY-er here with a ventilation question. Thanks in advance for your thoughts.

Context of the area and situation: We have ~13ft vaulted ceilings in our house in the living room and then a taller ~16ft vaulted ceiling in the entryway adjacent to the living room. These rooms are separated by a wall that opens through a doorway of about average height and width. In the summer hot air accumulates in both vaulted areas that are divided by this wall. In the entry ceiling we have our only HVAC intake vent. I can feel the difference when I stick my hands up into the air on both sides of the doorway, the living rooms is always noticeably hotter. The entry also has a skylight that we want to replace with an opening skylight to allow for additional ventilation and cooling.

Objective: I would love to add a passive vent (approximately 16x20in) high up in the dividing wall that would allow the hot air trapped in the vaulted living room to also vent out into the adjacent entry and be moved by the HVAC or vented through the open skylight. Theres no existing plumbing, electrical, or venting in this wall. If I simply make a hole in the wall, leaving existing studs, and cover with a grate, will that work? Or should I figure out removing the studs and framing the opening etc? Should I install ducting between the two sides of the wall? Any other considerations I should account for before cutting a giant hole in my wall?

Thank you so much for your thoughts on this unique project!

TLDR: I want to install a large passive air vent in a vertical wall straight through, do I need to frame it and add duct work for a single wall width?