The structure of the training set should be as follow:

...

In few words, the label of the class, and the value of (32 x 32 x 3) pixel, in order R values, G values and B Values. 

As well as CIFAR10 Classification Problem, the training set consists of 2D Data (in this case the first solution you wrote in the question title). 

A good reference for a baseline approach to CIFAR10:

https://www.cs.toronto.edu/~kriz/cifar.html

Hope this could be useful for your research.