The Dirac matrix have the following properties
$$ \gamma_\mu\gamma_\nu = \eta_{\mu\nu}\\ $$
This works in 3+1D because the signature is $(+,-,-,-)$.
They can be used to represent a vector in 3+1D like this $v = t\gamma_0+ x\gamma_1+y\gamma_2+z\gamma_3$.
In 3D, we have the Pauli matrices
$$ \sigma_i\sigma_j = \delta_{ij} $$
One can write a vector in 3D like this $v=x\gamma_x+y\gamma_y+z\gamma_z$.
I am looking for 4x4 matrices which works for signature $(+,+,+,+)$. The properties would be
$$ \mathbf{\hat{x}}_i \mathbf{\hat{x}}_j= \delta_{ij} $$
The matrices would be 4x4 and would allow one to write a vector $v=x \mathbf{\hat{x}} + y \mathbf{\hat{y}} + z \mathbf{\hat{z}} + w\mathbf{\hat{w}}$
Are these matrices known? I would like to know the entries of those matrices.