Sure. It doesn't have to be a full monthly layout.
Here's how to make a simple weekly chalkboard design:
Divide your board into 8 rows.
In the top row, add a header to the top of your sign with the date range of the week.
In row 2, write 'Monday' on the left (keep it small).
Keep going, row by row, with each day of the week.
Now you can add your calendar items in each row/day of the week!
TIP: If you want to be fancier, don't draw your rows the full width of the board to leave space for a few extra square sections for 'Notes' or 'Dinner Menu'.