Theoretically, yes, but not by lifting “straight” up. Simply being aloft does not divorce you from the forces that were acting on you before liftoff. When your helicopter was sitting on the launch pad, it was apparently motionless, right? But in reality, it was sitting on a globe whizzing around with a rotational speed of approximately 1,000 miles per hour.
So when you lift “straight” off, you are still subject to the same 1,000 mph rotational velocity that you were when on the ground. Without compensating for this, the result is a curving arc: slight motion in the Y direction as you lift off, but breakneck speed in the X direction.
This curving trajectory of an object launched from a rotating object is what’s known as the Coriolis effect, and is what makes launching a rocket “rocket science.” The rocket doesn’t just go straight up to its intended target (be it the moon, Mars, or Earth orbit), but instead – just like your helicopter – curves to the side as well.
So in order to stay “motionless” with respect to space, you’ve got to overcome that high-velocity send-off. To do so, you’d have to fly your helicopter in the opposite direction of Earth’s spin, which would give you a relative ground speed of 1,000 mph.
From the point of view of a motionless outside observer somewhere in space, you would then indeed seem stationary, but to you, inside the helicopter, it would seem like you’re cruising away at nearly twice the speed of sound!
Hope that helps.
~ Complacency is far more dangerous than outrage ~