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Let $n\geq 3$. One can show that the orthogonal group of degree $n$ over the real field, $\mathcal{O}(n,\mathbb R)$, spans the entire vector space of real $n\times n$ matrices, $\mathcal{M}(n,\mathbb R)$. More precisely if $k(n)$ denotes the smallest integer such that each $M\in \mathcal{M}(n,\mathbb R)$ can be written as $M=\sum_{i=1}^{k(n)} \lambda_i O_i, \quad \text{with}\quad (\lambda_i, O_i)\in (\mathbb R, \mathcal{O}(n,\mathbb R)).$ After showing that $\forall n>2, k(n)\leq 4$, can one find the integers such that $k(n)=4$?

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    The following paper may be helpful: Chi-Kwong Li and Edward Poon, *Additive Decomposition of Real Matrices*, Linear and Multilinear Algebra, 50(4):321-326, 2002. It essentially says that $k(n)\le 4$, but whether the bound is tight is an open problem.2011-10-27

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The following paper may be helpful: Chi-Kwong Li and Edward Poon, Additive Decomposition of Real Matrices, Linear and Multilinear Algebra, 50(4):321-326, 2002. It essentially says that $k(n)\le4$, but whether the bound is tight is an open problem.