Origin of the cross
Cross was used as a pagan symbol long before the advent of Jesus the Christ.
It had its origin in ancient Chaldea, used as the symbol of the pagan god Tammuz. It represented the shape of the mystic Tau which was the initials of the pagan god's name. (Vine’s Expository Dictionary of Old and New Testament Words)
Roman method of execution
During the time of Jesus, the Romans were using either an upright stake, also known as crux simplex pictured below, or a stake with a crossbeam which is simply called a cross as a method of execution, (New World Encyclopedia)
Crux simplex, a simple wooden torture stake, by Justus Lipsius.
The fact that the Romans were using an upright stake as a method of execution supports the Biblical account that Jesus was indeed impaled on an upright stake instead of a cross.
Note that both hands were nailed together above the head. In Jesus' case, the sign “Kings of the Jews” would have been placed right above the part where His hands were nailed, for everyone to see.
Thoughts to consider
Since a cross is a pagan symbol, would God the Father allow His Son to be impaled on a pagan object of worship?
The idea that Jesus carried a crossbeam on His way to Golgotha and that the post was already set up in Golgotha ready for reuse begs this question: Would God the Father allow the pure blood of His Son, which was meant to cleanse the sins of mankind, to be intermingled with the blood of criminals previously impaled?
Finally, doesn't the second commandment prohibit idolatry? Exodus 20:4-6
Exod 20:4 “You shall not make for yourself a carved image–any likeness of anything that is in heaven above, or that is in the earth beneath, or that is in the water under the earth;
5 you shall not bow down to them nor serve them. For I, the LORD your God, am a jealous God, visiting the iniquity of the fathers upon the children to the third and fourth generations of those who hate Me,
6 but showing mercy to thousands, to those who love Me and keep My commandments.
Would God the Father allow Jesus to be impaled on a cross which could be used as a symbol of worship and idolatry?
These are thought provoking questions which would only lead one to draw this conclusion: Jesus carried the staurus with the sign "The King of the Jews" to Golgotha as the Gospel accounts indicate, the very upright stake He was impaled upon. It's definitely not a cross.