No. You absolutely can see frames you generate or are destined for your radio without monitor mode. With the exception of the initial 802.1x handshake, unfortunately. You will need something like wireshark or tcpdump. However if you're talking about the 802.11 EAP you'll need a device you can use in monitor mode since they aren't passed through as standard frames: (might I suggest an ALFA AWUS036H)
They both rely on something called libpcap to allow you access to the layer2 messages your network devices see. Now this isn't perfect and the packets can get modified before you see them by your NIC. (eg. if you're capturing on a windows system, you'll likely see errors for TCP and IPv4 Checksum mismatches, that's due to a feature called TCP Offload Engine.) Monitor mode should get passed those limitations.
Unfortunately I'm not aware of any way to grab the pre-auth 802.1x packets without monitor mode. The first frame you'll see inside a pcap like this is likely the DHCP request frame.
What might work, is if you install a virtualization environment (VirtualBox or Vmware for instance) and then attempt to setup bridge mode on the interface and then use the virtual adapter to snoop the 802.1x frames. You will need to authenticate within your virtual machine but then you could presumably snoop it through the virtual interface.