Yes, the serpent in Genesis chapter 3 was Satan. Satan was either appearing as a serpent, possessing the serpent, or deceiving Adam and Eve into believing that it was the serpent who was talking to them. Serpents / snakes do not possess the ability to speak. Revelation 12:9 and 20:2 both describe Satan as a serpent. “He seized the dragon, that ancient serpent, who is the devil, or Satan, and bound him for a thousand years” (Revelation 20:2). “The great dragon was hurled down—that ancient serpent called the devil, or Satan, who leads the whole world astray. He was hurled to the earth, and his angels with him” (Revelation 12:9).
While the Bible is not clear as to whether or not the serpent stood up or walked before the curse, it appears likely that like other reptiles it probably did walk on four legs. That would seem to be the best explanation of Genesis 3:14, “So the LORD God said to the serpent, ‘Because you have done this, cursed are you above all the livestock and all the wild animals! You will crawl on your belly and you will eat dust all the days of your life.’” The fact that the serpent was cursed to crawl on his belly and eat the dust of the earth forever is also a way of indicating that the serpent would be forever despised and looked upon as a vile and despicable creature and an object of scorn and contempt. Why did God curse the serpent when He knew that it was actually Satan who had led Adam and Eve into sin? The fate of the serpent is an illustration. The curse of the serpent will one day be the fate of Satan himself (Revelation 20:10; Ezekiel 28:18-19).