Visualizing Support Vector Machines (SVM) with Multiple Explanatory Variables
Does it matter that the model is created in the form of SVM?

If no, I have seen a clever 6-D visualization. Its varieties are becoming popular in medical presentations. 
3 dimensions are shown as usual, in orthographic projection.
Dimension 4 is color (0..255)
Dimension 5 is thickness of the symbol 
Dimension 6 requires animation. It is a frequency of vibration of a dot on the screen. 
In static, printed versions, one can replace frequency of vibration by blur around the point, for a comparable visual perception.

If yes, and you specifically need to draw separating hyperplanes, and make them look like lines\planes, the previous trick will not produce good results. Multiple 3-D images are better.