scale-free networks and adjacency matrix
The most popular generative model for scale-free networks is the preferential attachment model of Barabási and Albert. A graph is sampled using an iterative procedure. This is a different sampling procedure than you suggest, since the goal is not only to generate a certain degree sequence, but also to mimic other properties of real-life networks.

If all you're after is a certain degree sequence, you can use MCMC methods; see for example Stanton and Pinar. Such algorithms don't conform to your type of algorithm, however. If you want to use your type of model, you have to decide in advance which vertices will have high degree and which will have low degree, since otherwise you will get a degree distribution which is either roughly Gaussian or roughly Poisson, but definitely no power law.

If all you're after is a certain degree sequence and you really want to use your own generative model, here is one possibility. Use the preferential attachment model, the algorithm of Stanton and Pinar, or any other randomized algorithm to generate a graph $G$. Now your model is: given two vertices $x,y$, if $(x,y) \in G$ then connect $x$ and $y$ with probability $1$, otherwise connect them with probability $0$.