Turn RPi3 Wi-Fi adapter into monitor mode using airmon-ng
You cannot. Not all chipsets/wifi drivers support monitor mode. Broadcom is known for lacking in open source drivers functionality support.
It is already public knowledge the RPi 3 current driver implementation does not support monitor mode.
PSA: The Raspberry Pi 3's embedded WiFi card does not support promiscuous mode.
If you are buying a replacement, do not go for a cheap realtek one, lots of bugs. Buy ralink or atheros, and check first if the chipset (and monitor mode) is well supported by the Linux kernel.
I am personally happy with this 8-10 Euro ralink, bought in Aliexpress: (link dead several times, replaced by the characteristics): 2T2R 300Mbps Dual Band 2.4GHz 5.8GHz-Ralink RT5572N WiFi USB-Adapter
see also https://unix.stackexchange.com/questions/252210/wi-fi-problems-using-asus-usb-n13-adapter-realtek
The default firmware does not allow monitor mode. The Nexmon project has patches to enable that.