I was reading [my professor's notes](https://drive.google.com/file/d/1Ts5AA92FIJ0_EaRRmNjmnVX2Qg_ZpAAa/view?usp=sharing) and [this answer](https://physics.stackexchange.com/a/269009/230829) on Dirac matrices.
Both sources reached the conclusion that for $N$-dimensional Clifford algebra, the basis is made up of $2^N$ independent matrices.

While the notes used $4$-dimensional matrices as an example, which have $4×4 =16$ elements(and makes it kinda intuitive that it is spanned by $2^4=16$ independent matrices), this argument falls for higher dimensions.

For example, $6$-dimensional matrices have 36 elements, but are spanned by $2^6=64$ **independent(?)** matrices.

This feels contradictory because the matrix $6×6$ has 36 elements, so there's maximally 36 independent matrices that form that basis. Am I correct?

PS: Linear algebra is not my strongest field, so I apologize if I'm missing a trivial point.