Mosfet keep gate information
With most Arduino outputs, when you do AnalogWrite you are not setting the voltage of the output.  You are setting the duty cycle (percent on-time) of a digital signal that turns on and off.  There are other chips that can generate that type of signal independent of the Arduino output but they are more complex to implement and program than just using another Arduino processor.

The Arduino Due (and most other ARM based Arduinos) have a couple of actual analog outputs (DACs) which changes what is happening with the gate, but doesn't really change the solution.

So, in short, no.  There is no simple way to keep the gate in that state after disconnecting the Arduino pin.  Depending on what you are actually trying to accomplish there might be other solutions.

EDIT - google Charlieplexing if you want to drive a large number of leds from an arduino.