To answer your question it's important to understand exactly what the FRW solution is. The GR field equations give you a way to calculate the metric given some distribution of matter and energy (aka the stress-energy tensor). If you assume the universe is full of matter that is homogeneous, isotropic and non-interacting (except for gravity), then feed this into the field equations they tell you that the metric describing this universe is the FRW metric.
The metric allows you to calculate the curvature of the universe locally i.e. if you take some point you can work out the curvature of the universe around you. It doesn't tell you global properties like the topology, so for example the FRW metric wouldn't tell you whether the universe is an infinite sheet or a torus. Both would be possible.
All we get from GR is the metric, but then this is all we need. The metric tells us the local curvature, and because we started with the assumption that the universe is homogeneous and isotropic we know that the curvature is the same everywhere. Note that the metric describes spacetime not just space. If you know the metric at some starting point you can use it to work out the evolution of the universe in time as well as space. In fact this is exactly how the big bang theory came to exist. The FRW metric predicts the universe must have started at a singularity, and that it will end either as another singularity or infinite expansion depending on the value of $\Omega$.
So to get back to your question, there is no "shape" that is distinct from the metric. The metric/curvature is all there is. I mentioned that the global topology could be anything, but I suspect this isn't what you meant by "shape". Incidentally, when I mention the topology could be a torus, that doesn't mean space is curved into a dougnut shape. The topology just describes the connectivity i.e. a torus just means that whatever direction you move in you'll eventually get back to where you started. The universe could be a torus and still be flat (I'm not sure if an $\Omega$ > 1 universe could be a torus).
Were you thinking about the topology when you mentioned a cone? The problem with a cone topology is that spacetime wouldn't be isotropic. There would be some directions (around the cone) where you get back to where you started, but other directions (away from the tip) where you wouldn't get back to where you started. Also there'd be a singularity at the tip.