I know I can do this by holding shift and then typing the letter of the axis I want to be excluded. However, I really like how I can scale on one axis by hitting "s" and then just use the mouse wheel to click and move in the direction I want to scale. However, I haven't found a way to exclude an axis using this method. Does anyone know how to do this?
You can't do it with the mouse only, but after pressing S, while the scaling operator is active you can press Middle Mouse Button to constraint to an axis, depending on the direction the mouse is pointing.
Similarly you can also press ⇧ Shift + Middle Mouse Button to exclude the same axis.