In fact, the Old Testament is best described eschatological in a historic setting. The reason that we don't recognize it, is that it is an eschatology that looks forward to something that (for the most part) has now been fulfilled: primarly the first advent and, I believe, the inauguration of the Kingdom of God.
In the "faith" chapter of Hebrews, chapter 11, we read:
All these people were still living by faith when they died. They did not receive the things promised; they only saw them and welcomed them from a distance. And they admitted that they were aliens and strangers on earth. (Hebrews 11:13)"These people" refers to the patriarchs, Abraham, Isaac, Jacob, etc., as well as other great practitioners of faith, such as Rahab. The lived by faith in eschatological promises never fulfilled in their lifetime, such as the occupation of the Promised Land and the coming of the Messiah. Likewise we live with a promise, most likely to remain unfulfilled in our lifetime, of the Second Advent.
As early as Genesis 3 we find an eschatological promise:
And I will put enmity between you and the woman, and between your offspring and hers; he will crush your head, and you will strike his heel." (Gen 3:15)Here God promises a Redeemer to crush Satan’s head, which is accomplished through Christ’s finished work on the Cross.
The ancestry of the promised Redeemer is refined throughout the Old Testament. He is the seed of Eve in Gen 3:15, a descendant of Abraham in Gen 22:18, of the tribe of Judah in Gen 49:10, and the descendent of David in 2 Sam 7:12-13.
Another eschatological promise in the Old Testament is the New Covenant. We read in Jeremiah:
31 "The time is coming," declares the LORD , "when I will make a new covenant with the house of Israel and with the house of Judah. 32 It will not be like the covenant I made with their forefathers when I took them by the hand to lead them out of Egypt, because they broke my covenant, though I was a husband to them, " declares the LORD . (Jer. 31:31,32)In Hebrews 8:8-13 it is confirmed that this New Covenant was ushered in by Christ.
Other Old Testament eschatological promises include: The restoration of Israel (Jer. 23:3) (although exactly what that means is the subject of intense debate), The Day of the Lord (e.g., Isa. 13:9-11, Zeph 1:14-15) (again, fierce debate as to what that means) and Pentecost (Joel 2:28-19).
There is even Old Testament eschatological promise for which there is nearly universal agreement (save for full preterism proponents) that it has not yet been fulfilled: A new heavens and earth:
"Behold, I will create new heavens and a new earth. The former things will not be remembered, nor will they come to mind. (Isa. 65:17).
Do not mistakenly view that Old Testament as having been written as a journal, merely recording events as they occurred. It is overwhelmingly forward-looking.