(Not a teacher)
Consider "Game over" to be a mass noun, and use it as such. I would say the etymology of the word as a whole came from arcade games; I doubt it was used before then as a single noun.
You can say "the game is over", but "game over" itself means that something has ended unsucessfully, and can be used outwith gaming meaning that the game is finished. For example, "If I don't pass this exam it's game over for me".