For chrome to rust that quick, the plating basically sucks.
Good chrome on steel starts with a highly polished and degreased steel, then it is copper plated - moderate thickness and polished again. Then it is nickel plated - thickly, and polished yet again. The chrome plating is then pretty thin as one gets the best shine with a thin layer.
Chrome itself is not waterproof, hence the need for a good thick nickel layer.
You can't tell quality plating by looking at it, smear with salt and leave wet and if it's poor, it'll rust in days!
