Did Draco Malfoy take the elder wand from Albus Dumbledore after he died?

Draco was the actual master of the elder wand because Dumbledore's death was already planned, so when Snape killed him, it wasn't actually a takeover of the wand. Because Draco was the last to disarm Dumbledore, or 'take' his wand, it recognized Draco as it's master. This is how Voldemort was ultimately killed, because Harry was using Draco's original wand. The elder wand recognized the wand as its master's, so Voldemort's killing curse was turned back upon him.

However, Draco never actually had the Elder Wand in his possession. He never even knew he was the master until Harry revealed it.