Yes, any chart that spans the horizon may be used, and all will make identical physical predictions. Peter has suggested a convenient coordinate chart to use and provided a link (for the idealized non rotating BH).The point of view you choose shouldn't affect the reality of the physics, should it? I mean even if different observers have different perspectives there should be some calculable "reality" as to what happens which all should agree on when spacetime distortions are considered. Can't there be one chart in some sense that can be translated into any reference frame or is that physically unrealistic? (neglecting quantum effects)

Another less commonly used chart that I happen to like is the Lemaitre chart:

https://en.wikipedia.org/wiki/Lemaître_coordinates

These have the feature of maintaining 1 timelike and 3 spacelike coordinates throughout the exterior and interior (Gullestrand-Panlieve coordinates are all spacelike inside the horizon). Kruskal coordinates also maintain 1 timelike and 3 spacelike coordinates everywhere, but I find them harder for many computations. In Lemaitre coordinates, free fall trajectories from infinity have a very simple representation, and the time coordinate gives proper time along such trajectories.

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