Generating n random variables, and multiplying them together c++

For the sake of simplicity, let's say the user enters n = 2, what I want to happen is for two (or n) random numbers to be multiplied together. It should work for any n, and where the random numbers are between 0 and 1.

I have been using this for random numbers between 0 and 1
(double)rand()/(double)RAND_MAX for this.

So to break it down further
n= 1
random_number = random_number 
n = 2
random_number = random_number * random_number
n = 3
random_number = random_number * random_number * random_number 
n = ...
random_number = random_number * random_number * ...

The problem is the random numbers have to be unique as well. For example, for the case of n = 2, it should be ...

 random_number = (unique)random_number * (unique)random_number

To complicate the matter further, let's also say I have m (not static) divisions, and I create a vector of size m^n. I also need to create this vector of size m^n to be...

 vector = [rand*rand, rand*rand, rand*rand, rand*rand]
 for the case when n = 2 m = 2

How would I go about doing this in c++?