Is it true?
Electromagnetic fiel tensor, defined by:
$\mathbf{F}\overset{\text{def}}=B+E\wedge \mathrm{d}t= B_1\mathrm{d}y\wedge \mathrm{d}z+ B_2\mathrm{d}z\wedge \mathrm{d}x+ B_3\mathrm{d}x\wedge \mathrm{d}y-\frac{E_1}{c}c\mathrm{d}t\wedge \mathrm{d}x -\frac{E_2}{c}c\mathrm{d}t\wedge \mathrm{d}y -\frac{E_3}{c}c\mathrm{d}t\wedge \mathrm{d}z$
is (usualy) express using this matrix (first row and second column):
$F_{\mu\nu}=\left(\begin{array}{cccc} \ 0&E_1/c&E_2/c&E_3/c\\ -E_1/c&0&\color{red}-B_3&\,\,B_2\\ -E_2/c&\,\,B_3&0&\color{red}-B_1\\ -E_3/c& \color{red}-B_2&\,\,B_1&\ 0\end{array}\right)$
in ortonormal basis: $\Big(\mathrm{d}x\wedge c\mathrm{d}t,\, \mathrm{d}y\wedge c\mathrm{d}t,\, \mathrm{d}z\wedge c\mathrm{d}t,\, \mathrm{d}y\wedge\mathrm{d}z,\, \mathrm{d}z\wedge\mathrm{d}x,\, \mathrm{d}x\wedge\mathrm{d}y\Big)$,
where $c$ in SI units ensures orthonormality and Minkowski metric tensor is: $\small\eta= \begin{pmatrix} c\mathrm{d}t& \mathrm{d}x& \mathrm{d}y& \mathrm{d}z\end{pmatrix} \begin{pmatrix}-1&0&0&0\\ 0&1&0&0\\ 0&0&1&0\\ 0&0&0&1 \end{pmatrix} \begin{pmatrix} c\mathrm{d}t\\ \mathrm{d}x\\ \mathrm{d}y\\ \mathrm{d}z\end{pmatrix}$.
Is it posible tensor $F$ express in (orthonormal) basis using matrix multiplication like ordinary vector or covector? How? Thanks.