A Lie group G is said to be uniformly finitely generated by one-parameter subgroups exp(tXi), i = 1, . . . , n, if there exists a positive integer k such that every element of G may be expressed as a product of at most k elements chosen alternatively from these one-parameter subgroups. In this paper we construct sets of left invariant vector fields on SO(n), in particular, pairs {A, B}, whose one-parameter subgroups uniformly finitely generate SO(n) and find an upper bound on the order of generation of SO(n,R) by these subgroups. We give special attention to the case n = 3. 0. Introduction. If the Lie algebra of a connected Lie group G is generated by the elements X1, . . . , Xn, then every element of G may be expressed as a finite product of elements of the form exp(tXi), where t is real and i = 1, . . . , n (Jurdjevic and Sussmann [6]). However, the number of elements required for g ∈ G may not be uniformly bounded as g ranges through G. If, in addition, G is compact and exp(tXi), i = 1, . . . , n are also compact, then it follows from Theorem 1.1 that there exists a positive integer k such that every element of G may be expressed as a product of at most k elements from exp(tXi), i = 1, . . . , n. That is, G is uniformly finitely generated by these oneparameter subgroups with order of generation k. For two and three-dimensional Lie groups, the problem has been completely solved by Koch and Lowenthal. In [1], Crouch and the present author take the initial steps in the problem of uniform finite generation of SO(n,R) (the real n(n− 1)/2-dimensional special orthogonal group with Lie algebra so(n)) and concentrate on finding pairs of generators for so(n), orthogonal with respect to the killing form 〈·, ·〉 and whose one-parameter subgroups uniformly finitely generate SO(n). This paper is still devoted to the uniform generation problem of SO(n). Section 1 is introductory. Sections 2 and 3 are concerned with Work supported in part by Centro de Matematica da Universidade de CoimbraINIC and by JNICT under project 87.62. Received by the editors on May 22, 1984 and in revised form on May 27, 1988. Copyright c ©1991 Rocky Mountain Mathematics Consortium