Teresa,

Okay, here we go . Your Absalom Skipworth served in Company B of Hardeman's ( Col Willaim P.) Regiment , which was known as the 1st Cavalry Regiment of the Arizona Brigade. They were also known as the 31st Texas Cavalry Regiment . However, they ( Hardeman's men) should not be confused with Hawpe's 31st Texas Cavalry Regiment !! That was a different outfit all together.
Hardeman's Cavalry Regiment was organized by adding Hardeman's Cavalry Battalion , Arizona Brigade to another regiment on Feb.21,1863 . Hardeman's Cavalry Regiment was dismounted in March 1865 and was surrendered by Edmund Kirby Smith on May 26, 1865. Along the way, it looks like your man participated in the Camden Expedition ( March - May 1864) , fought in the battles of Poison Spring ( 4/18/64) , Massard's Prairie ( 7/27/64) and Cabin Creek ( Indian Territory , aka Oklahoma) on September 19, 1864.
Also, the National Park Service Civil War Soldiers and Sailors System , lists him as entering service as a private and mustering out as a private in Co. B.

Probably the most noteworthy of these battles listed above is that of the Confederate victory at Cabin Creek. Serving under General Gano's overall command, Private Skipworth's outfit helped defeat a Federal force and capture a large supply wagon train , whose contents were valued at $ two million US dollars. More importantly, the supply train contained food, clothing, weapons, and ammunition; all items that were sorely needed not only by Gano's men but also by the starving Confederate civilian refugees who were encamped along Boggy Depot in southern Indian Territory . This victory was probably the last good bit of fortune these folks received until the fighting stopped in May of 1865.
Hope this helps you in your hunt .
Best Regards, Craig Morin