Question: How do I generate all 4x4 rank-1 matrices over the field F_2?

I want to generate all 4x4 matrices of rank-1 over the field with two elements, F_2. I don't want to use actual arrays though, I just want to use symbols representing the unit matrices. So for example Eij represents the 4x4 matrix with a 1 in row i & column j, and 0's elsewhere. 

We have 16 unit matrices: E11, E12, ..., E14, ... E41, E42, E43, E44. Then we have to look at all possible linear combinations that result in a rank-1 matrix. So for example: E11+E12,  E11+E21,  E11+E12+E13,  E33+E43, etc. The total number of rank-1 4x4 matrices over F_2 is 225. How can I find all of these quickly?

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