Lt. Col. Seth "Jethro" Nehring and Ukrainian pilot Col. Ivan Petrenko were killed in the crash.
The airman was a member of the California Air National Guard's 144th Fighter Wing, U.S. Air Forces Europe-Africa said.
The pilot was reportedly aboard the Ukrainian fighter when it crashed.
Russia has already condemned the Trident Juncture exercise as a needless provocation that "hasn't gone unnoticed."
Navy combat photographers and cameramen have documented every major U.S. conflict since World War I.
Three U.S. soldiers were injured when their Abrams tank struck an embankment Wednesday during an exercise in Slovakia.
Announcement of the Clear Sky exercise comes days after Ukraine said it would establish a naval base along the Sea of Azov.
Russia and China intend to regularly conduct joint war games similar to the massive ones being held this week.
Russia launched its largest military exercise since the Cold War, war games that will involve thousands of Chinese troops.
The U.S. military has scheduled its exercises in the Gulf of Alaska for spring, despite calls for them to be moved to fall.