It's more probable you've stumbled into a cave. Was the hole on a cliff? Mine holes themselves are player made, and appear as holes dug in the earth with a rope ladder descending into them.
You could still be in luck, with a stone axe in your hand walk around the cave using the mining skill on the walls (you need to be a little ways away from the entrance for this to work, FYI.) depending on your strength you may have to look around a little before you can find anything soft enough to mine. Once you successfully begin mining a wall segment, you will either see stones dropping to your feet, or you will have ore appear in your inventory. Mining takes an absurd amount of stamina, especially when you are first starting out and don't have a metal pick so be sure to bring a bucket of water and a snack.
As a last few pieces of advice, be very careful and try not to mine out consecutive spaces. Digging two next to each other is okay if you've found ore, but I would not recommend a third as its quite likely to cause a cave in. Also, All ore appears to label itself as Iron, so if you do luck out and find an ore seam: Smelt it up and see what kinda presents you found

Edit: Also forgot to mention that ore is located in seams. Meaning if you find a lucky spot on the wall you just found yourself a "free" mine
