For me, don’t rush things yet. Like you said, you just started seeing each other last month, so give it more time to really get to know each other.
Maybe wait for him to be the one to open up first and ask if you can be exclusive, since he’s the guy and he might take the lead on that.
For now, just go with the flow and enjoy your “getting-to-know-each-other” stage and the mutual understanding you’re building.