How can we know the status of the DC Motor whether it is ON or OFF
Unfortunately, the L298N does not return a status, so you are going to have to add sensors to detect motor movement.
Here are a few options you can try:

Current sense. Add a small resistor to the output of the L298N and measure the voltage differential across that resistor. If there is current being supplied to the motor then you will have a voltage differential.
Rotation sensor. This can be in the form of a quadrature encoder wheel, tooth sensor or Hall effect sensor detecting a magnet on the shaft. If you see pulses from the sensor then you know the motor is spinning.
Optical sensor. A camera system or other optical sensor to detect motor movement. Probably too complex for a simple application. There are however off the shelf sensors that will turn on an output if they see movement in their field of view. Alternatively you could use a PIR sensor from a home alarm system if the movement is course enough.